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41 results on '"Mario Ricchiuto"'

1. Adaptive deformation of 3D unstructured meshes with curved body fitted boundaries with application to unsteady compressible flows

Author

Barbara Re, Alberto Guardone, Giuseppe Quaranta, Mario Ricchiuto, Algiane Froehly, Luca Cirrottola, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Direction générale déléguée à l'innovation (DGD-I), Institut National de Recherche en Informatique et en Automatique (Inria), Institute of Mathematics University of Zurich, Department of Aerospace Science and Technology, Politecnico di Milano [Milan] (POLIMI), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Physics and Astronomy (miscellaneous), Computer science, Boundary (topology), Unstructured meshes, 010103 numerical & computational mathematics, Slip (materials science), Deformation (meteorology), 01 natural sciences, Domain (mathematical analysis), [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Polygon mesh, Conservative formulations, Constant-connectivity mesh adaptation, Unsteady compressible flows, 0101 mathematics, Constant-connectiity mesh adaptation, Numerical Analysis, Applied Mathematics, Mathematical analysis, [INFO.INFO-CE]Computer Science [cs]/Computational Engineering, Finance, and Science [cs.CE], [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Compressibility, Gravitational singularity, Laplace operator
Abstract

International audience; We present an adaptive moving mesh method for unstructured meshes which is a threedimensional extension of the previous works of Ceniceros et al. Ceniceros2001 [9], Tang et al. Tang2003 [38] and Chen et al. Chen2008 [10]. The iterative solution of a variable diffusion Laplacian model on the reference domain is used to adapt the mesh to moving sharp solution fronts while imposing slip conditions for the displacements on curved boundary surfaces. To this aim, we present an approach to project the nodes on a given curved geometry, as well as an a-posteriori limiter for the nodal displacements developed to guarantee the validity of the adapted mesh also over non-convex curved boundaries with singularities. We validate the method on analytical test cases, and we show its application to two and three-dimensional unsteady compressible flows by coupling it to a second order conservative Arbitrary Lagrangian-Eulerian flow solver.

Published
2021

2. A spectral/hpelement depth-integrated model for nonlinear wave–body interaction

Author

Mario Ricchiuto, Claes Eskilsson, Allan Peter Engsig-Karup, Umberto Bosi, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Danmarks Tekniske Universitet = Technical University of Denmark (DTU), RISE Research Institutes of Sweden, Ocean ERANET project MIDWEST, ADEME, SWEA, FCT, PLAFRIM, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Technical University of Denmark [Lyngby] (DTU)

Subjects
Discretization, Computational Mechanics, General Physics and Astronomy, 010103 numerical & computational mathematics, 01 natural sciences, Wave-body interaction, Nonlinear and dispersive waves, Discontinuous Galerkin method, Wave–body interaction, Spectral/hp element method, Convergence (routing), [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Domain decomposition, 0101 mathematics, Boussinesq equations, Spectral/ hp element method, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Coupling, Physics, Mechanical Engineering, Mathematical analysis, Domain decomposition methods, [SPI.GCIV.CH]Engineering Sciences [physics]/Civil Engineering/Construction hydraulique, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, Exponential function, 010101 applied mathematics, Nonlinear system, Flow (mathematics), Mechanics of Materials, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We present a spectral/hp element method for a depth-integrated Boussinsq model for the efficient simulation of nonlinear wave-body interaction. The model exploits a `unified' Boussinesq framework, i.e. the flow under the body is also treated with the depth-integrated approach, initially proposed by (Jiang, 2001) and more recently rigorously analysed by (Lannes, 2016). The choice of the Boussinesq equations allows the elimination of the vertical dimension, resulting in a wave-body model with an adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. In this work we employ flux-based conditions for domain coupling, following the recipes provided by the discontinuous Galerkin spectral/hp element framework. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model. The spectral/hp element method allows to achieve high-order, possibly exponential, convergence for non-breaking waves and account for the nonlinear interaction with fixed and floating bodies. Our main contribution is to include floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is validated against published results for wave-body interaction and confirmed to have excellent accuracy. The proposed nonlinear model is demonstrated to be relevant for the simulation of wave energy devices.

Published
2019

3. Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping

Author

Rémi Abgrall, Sixtine Michel, Davide Torlo, Mario Ricchiuto, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Universität Zürich [Zürich] = University of Zurich (UZH), University of Zurich, Michel, Sixtine, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Discretization, Continuous Interior Penalty, 340 Law, 610 Medicine & health, Fekete nodes, 010103 numerical & computational mathematics, 01 natural sciences, Theoretical Computer Science, Mathematics::Numerical Analysis, 510 Mathematics, 2604 Applied Mathematics, cubature nodes, FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, 2614 Theoretical Computer Science, 2612 Numerical Analysis, Mathematics, Numerical Analysis, Partial differential equation, Spectral element method, Applied Mathematics, Numerical analysis, Courant–Friedrichs–Lewy condition, General Engineering, Numerical Analysis (math.NA), Deferred Correction scheme MSC: 65M60, Mass matrix, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Bernstein polynomial, Continuous Galerkin method, Finite element method, Quadrature (mathematics), 1712 Software, 010101 applied mathematics, 10123 Institute of Mathematics, Computational Mathematics, Computational Theory and Mathematics, Dispersion analysis, 2200 General Engineering, Streamline Upwind Petrov-Galerkin, 2605 Computational Mathematics, Local Projection Stabilization, Software, 1703 Computational Theory and Mathematics, 65M60
Abstract

International audience; We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

Published
2021
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4. UnDiFi-2D: an Unstructured Discontinuity Fitting code for 2D grids

Author

Lorenzo Campoli, Mario Ricchiuto, Aldo Bonfiglioli, Alessia Assonitis, Mirco Ciallella, Renato Paciorri, Saint Petersburg State University (SPBU), Dipartimento di Ingegneria Meccanica e Aerospaziale [Roma La Sapienza] (DIMA), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Università degli studi della Basilicata [Potenza] (UNIBAS), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Unstructured grids, Computer science, Fortran, Shock-capturing, General Physics and Astronomy, FOS: Physical sciences, 010103 numerical & computational mathematics, computer.software_genre, 01 natural sciences, 010305 fluids & plasmas, Software, 0103 physical sciences, 0101 mathematics, computer.programming_language, business.industry, Programming language, Code reuse, Software development, Fluid Dynamics (physics.flu-dyn), Usability, Physics - Fluid Dynamics, Solver, Computational Physics (physics.comp-ph), CFD, fortran, shock-capturing, shock-fitting, unstructured grids, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Shock-fitting, Test case, Hardware and Architecture, business, computer, Physics - Computational Physics, Software versioning
Abstract

UnDiFi-2D, an open source (free software) Unstructured-grid, Discontinuity Fitting code, is presented. The aim of UnDiFi-2D is to model gas-dynamic discontinuities in two-dimensional (2D) flows as if they were true discontinuities of null thickness that bound regions of the flow-field where a smooth solution to the governing PDEs exists. UnDiFi-2D therefore needs to be coupled with an unstructured CFD solver that is used to discretize the governing PDEs within the smooth regions of the flow-field. Two different, in-house developed, CFD solvers are also included in the current distribution. The main features of the UnDiFi-2D software can be summarized as follows: Programming Language: UnDiFi-2D is written in standard Fortran 77/95; its design is highly modular in order to enhance simplicity of use, maintenance and allow coupling with virtually any existing CFD solver; Usability, Maintenance and Enhancement: In order to improve the usability, maintenance and enhancement of the code also the documentation has been carefully taken into account. The git distributed versioning system has been adopted to facilitate collaborative maintenance and code development; Copyrights: UnDiFi-2D is a free software that anyone can use, copy, distribute, change and improve under the GNU Public License version 3. The present paper is a manifesto of the first public release of the UnDiFi-2D code. It describes the currently implemented features, which are the result of more than a decade of still ongoing CFD developments. This work is focused on the computational techniques adopted and a detailed description of the main characteristics is reported. UnDiFi-2D capabilities are demonstrated by means of examples test cases. The design of the code allows to easily include existing CFD codes and is aimed at ease code reuse and readability., Comment: Keywords: CFD, Fortran, C, shock-fitting, shock-capturing,, unstructured grids

Published
2021
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5. Extrapolated Shock Tracking: bridging shock-fitting and embedded boundary methods

Author

Mario Ricchiuto, Mirco Ciallella, Renato Paciorri, Aldo Bonfiglioli, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], Università degli studi della Basilicata [Potenza] (UNIBAS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA)

Subjects
Shock wave, [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Bridging (networking), Physics and Astronomy (miscellaneous), Computer science, Phase (waves), Boundary (topology), Tracking (particle physics), 01 natural sciences, 010305 fluids & plasmas, Embedded-boundaries, embedded-boundary, shock-fitting, unstructured-grids, 0103 physical sciences, 0101 mathematics, Flexibility (engineering), Numerical Analysis, Applied Mathematics, Numerical analysis, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Shock-fitting, Computer Science Applications, Shock (mechanics), 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Algorithm, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose a novel approach to approximate numerically shock waves. The method combines the unstructured shock-fitting approach developed in the last decade by some of the authors, with ideas coming from embedded boundary techniques. The numerical method obtained allows avoiding the re-meshing phase required by the unstructured fitting method, while guaranteeing accuracy properties very close to those of the fitting approach. This new method has many similarities with front tracking approaches, and paves the way to shock-tracking techniques truly independent on the data and mesh structure used by the flow solver. The approach is tested on several problems showing accuracy properties very close to those of more expensive fitting methods, with a considerable gain in flexibility and generality.

Published
2020

6. Space-time residual distribution on moving meshes

Author

Matthew E. Hubbard, D. Sármány, Mario Ricchiuto, School of Mathematical Sciences [Nottingham], University of Nottingham, UK (UON), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), European Centre for Medium-Range Weather Forecasts (ECMWF), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Conservative ALE, 010103 numerical & computational mathematics, 01 natural sciences, Shallow water equations, Triangle mesh, Applied mathematics, Polygon mesh, Discontinuous space-time representation, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Mathematics, ComputingMethodologies_COMPUTERGRAPHICS, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Conservation law, Moving meshes, Space time, Mass balance, Order of accuracy, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, Upwind residual distribution, Well-balanced schemes, Laplace operator, Smoothing, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; This article investigates the potential for an r-adaptation algorithm to improve the efficiency of space-time residual distribution schemes in the approximation of time-dependent hyperbolic conservation laws, e.g. scalar advection, shallow water flows, on unstructured, triangular meshes. In this adaptive framework the connectivity of the mesh, and hence the number of degrees of freedom, remain fixed, but the mesh nodes are continually "relo-cated" as the flow evolves so that features of interest remain resolved as they move within the domain. Adaptive strategies of this type are well suited to the space-time residual distribution framework because, when the discrete representation is allowed to be discontinuous in time, these algorithms can be designed to be positive (and hence stable) for any choice of time-step, even on the distorted space-time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the time-step, which can instead be chosen according to accuracy requirements. The order of accuracy of the fixed-mesh scheme is retained on the moving mesh in the majority of applications tested. Space-time schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law, so moving the mesh does not artificially change the flow volume for pure conservation laws. For shallow water flows over variable bed topography, the so-called C-property (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved. The r-adaptation is applied within each time-step by interleaving the iterations of the nonlinear solver with updates to mesh node positions. The node movement is driven by a monitor function based on weighted approximations of the scaled gradient and Laplacian of the local solution and regularised by a smoothing iteration. Numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed * *Manuscript Click here to view linked References which show that, even for this simple implementation of the mesh movement, reductions in cpu times of up to 60% can be attained without increasing the error.

Published
2020

7. Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes

Author

Luca Arpaia, Mario Ricchiuto, Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), PLAFRIM, ANR-11-RSNR-0023,TANDEM,Tsunamis en Atlantique et MaNche : Définition des Effets par Modélisation(2011), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Moving Mesh, Physics and Astronomy (miscellaneous), Shallow Water equations, Well-Balanced, Context (language use), 010103 numerical & computational mathematics, 01 natural sciences, Spherical geometry, Arbitrary Lagrangian Eulerian formulation, Applied mathematics, Polygon mesh, 14. Life underwater, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Shallow water equations, Residual Distribution, Mathematics, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Numerical Analysis, Curvilinear coordinates, curvilinear coordinates, Applied Mathematics, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Manifold, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Transformation (function), Flow (mathematics), Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We consider the numerical approximation of the Shallow Water Equations (SWEs) in spherical geometry for oceanographic applications. To provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere. To this end, we restate all Arbitrary Lagrangian Eulerian (ALE) transport formulas, as well as the volume transformation laws, for a 2D manifold. Using these results, we write the set of ALE-SWEs on the sphere. We then propose a Residual Distribution discrete approximation of the governing equations. Classical properties as the DGCL and the C-property (well balancedness) are reformulated in this more general context. An adaptive mesh movement strategy is proposed. The discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement.

Published
2020

8. Residual based method for sediment transport

Author

Mario Ricchiuto, Priscilla Ramsamy, Pascal Poullet, Laboratoire de Mathématiques Informatique et Applications (LAMIA), Université des Antilles (UA), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Mathématiques Informatique et Applications [UR1_1] (LAMIA), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Sediment, 010103 numerical & computational mathematics, Residual, 01 natural sciences, 010101 applied mathematics, Runge–Kutta methods, Test case, Fluid dynamics, Limiter, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Flux limiter, 0101 mathematics, Sediment transport, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; This contribution deals with a high order Residual Distribution (RD) numerical scheme to simulate sediment transport. The morphodynamic model that has been used, couples shallow-water equations for the fluid flow and the Exner law for the sediment part. Thus, the choice of the approach by a non-conservative hyperbolic system has been made. Different schemes have already been applied to approximate the entropic solution for several test cases. The one proposed in this paper resorts to RD-method, TVD Runge Kutta and stabilisation upwind methods, with limiters. It can be viewed as an improvement of the generalized approximate Roe method with some other good properties (Path-conservative, well-balanced...). Numerical results show the ability of the model in 1D to compute accurate solutions and to reproduce some classical test problems. The best results that we obtained, use MinMod flux limiters.

Published
2020

9. An efficient covariant frame for the spherical shallow water equations: Well balanced DG approximation and application to tsunami and storm surge

Author

Rodrigo Pedreros, Andrea Gilberto Filippini, Mario Ricchiuto, Luca Arpaia, Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Atmospheric Science, 010504 meteorology & atmospheric sciences, Discretization, Hydrostatic pressure, 010103 numerical & computational mathematics, Discontinuous galerkin, Oceanography, 01 natural sciences, Projection (linear algebra), Shallow water equations, symbols.namesake, Discontinuous Galerkin method, Computer Science (miscellaneous), Covariant transformation, Tensor, 0101 mathematics, 0105 earth and related environmental sciences, Mathematics, Tsunami, Well balanced schemes, Mathematical analysis, Spherical geometry, Geotechnical Engineering and Engineering Geology, Storm surge, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Riemann problem, symbols
Abstract

International audience; In this work we consider an e cient discretization of the Shallow Water Equations in spherical geometry for oceanographic applications. Instead of the classical 2d-covariant or 3d-Cartesian approaches, we focus on the mixed 3d/2d form of [Bernard et al., JCP 2009] which evolves the 2d momentum tangential to the sphere by projecting the 3d-Cartesian right-hand side on the sphere surface. Di↵erently from the last reference we consider the exact representation of the sphere instead of the finite element one, mixed with a covariant basis projection of the momentum. This leads to several simplifications of the Discontinuous Galerkin scheme: the local mass matrix goes back to the standard block-diagonal form; the Riemann Problem does not require any tensor or vector rotations to align the bases on the two sides of an edge. Second we consider well balancing corrections related to relevant equilibrium states for tsunami and storm surge simulations. These corrections allow to compensate for the inherent non-exactness of the quadrature induced by the non-polynomial nature of both the geometrical mapping and of the covariant basis. In other words, these corrections are the order of the cubature error. We show that their addition makes the scheme exactly well balanced, and is is equivalent to recasting the integral of the hydrostatic pressure term in strong form. The proposed method is validated on academic benchmarks involving both smooth and discontinuous solutions, and applied to realistic tsunami and an historical storm surge simulation.

Published
2022

10. Dispersive and dispersive-like bores in channels with sloping banks

Author

Philippe Bonneton, Andrea Gilberto Filippini, Mario Ricchiuto, Rémi Chassagne, Université Grenoble Alpes (UGA), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Erosion torrentielle neige et avalanches (UR ETGR (ETNA)), Environnements et Paléoenvironnements OCéaniques (EPOC), Observatoire aquitain des sciences de l'univers (OASU), Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE)-Centre National de la Recherche Scientifique (CNRS), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Work partially funded by the TANDEM contract, reference ANR-11-RSNR-0023-01 of the French Programme Investissements d’Avenir, and by the project 'vagues extrêmes' funded by the Région Nouvelle Aquitaine (reference 2017-1R20107)., PLAFRIM, ANR-11-RSNR-0023-01,TANDEM,Tsunamis in the Atlantic and the English ChaNnel: Definition of the Effects through numerical Modeling(2014), Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), ANR-11-RSNR-0023,TANDEM,Tsunamis en Atlantique et MaNche : Définition des Effets par Modélisation(2011), Université Sciences et Technologies - Bordeaux 1 (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1 (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École Pratique des Hautes Études (EPHE), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Asymptotic analysis, 010504 meteorology & atmospheric sciences, Edge (geometry), 01 natural sciences, law.invention, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], symbols.namesake, law, Froude number, Kondratiev wave, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [SDU.STU.HY]Sciences of the Universe [physics]/Earth Sciences/Hydrology, [SDU.STU.OC]Sciences of the Universe [physics]/Earth Sciences/Oceanography, 0105 earth and related environmental sciences, Physics, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, geography, geography.geographical_feature_category, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Mechanical Engineering, Mechanics, Condensed Matter Physics, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Open-channel flow, 010101 applied mathematics, Undular bore, Tidal bore, Mechanics of Materials, symbols, Hydrostatic equilibrium
Abstract

[Departement_IRSTEA]Eaux [TR1_IRSTEA]RIVAGE; International audience; In this paper, a detailed analysis of undular bore dynamics in channels of variable cross-section is presented. Two undular bore regimes, LowFroude Number (LFN) and High Froude Number (HFN), are simulated with a Serre-Green-Naghdi model, and the results are compared with theexperiments by Treske. We show that contrary to Favre waves and HFN bores, which are controlled by dispersive non-hydrostatic mechanisms, LFN bores correspond to a hydrostatic phenomenon. The dispersive-like properties of the LFN bores is related to wave refraction on the banks in a way similar to that of edge waves in the nearshore. A fully hydrostatic asymptotic model for these dispersive-like bores is derived and compared to the observations, confirming our claim

Published
2019

11. Adaptive Moving Mesh Upwind Scheme for the Two-Species Chemotaxis Model

Author

Tong Wu, Mario Ricchiuto, Alina Chertock, Alexander Kurganov, Department of mathematics [North Carolina], North Carolina State University [Raleigh] (NC State), University of North Carolina System (UNC)-University of North Carolina System (UNC), Southern University of Science and Technology (SUSTech), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Southern University of Science and Technology of China (SUSTech), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Pattern formation, Upwind scheme, 010103 numerical & computational mathematics, 01 natural sciences, Finite- volume upwind method, Quantitative Biology::Cell Behavior, law.invention, law, Adaptive moving mesh (AMM) method, Cell density, AMS subject classification: 65M50, 65M08, 92C17, 35K57, 35M10, Cartesian coordinate system, 0101 mathematics, Singular (spiky) solutions, Mathematics, Numerical analysis, Small number, Chemotaxis, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, Two-species chemotaxis system, Common property, Biological system, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacteria/cells in response to an external stimulus, usually a chemical one. A common property of all chemotaxis systems is their ability to model a concentration phenomenon-rapid growth of the cell density in small neighborhoods of concentration points/curves. More precisely, the solution may develop singular, spiky structures, or even blow up in finite time. Therefore, the development of accurate and computationally efficient numerical methods for the chemotaxis models is a challenging task. We study the two-species Patlak-Keller-Segel type chemotaxis system, in which the two species do not compete, but have different chemotactic sensitivities, which may lead to a significantly difference in cell density growth rates. This phenomenon was numerically investigated in [Kurganov and Lukáčová-Medvid'ová, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 131-152] and [Chertock et al., Adv. Comput. Math., 44 (2018), pp. 327-350], where second-and higher-order methods on uniform Cartesian grids were developed. However, in order to achieve high resolution of the density spikes developed by the species with a lower chemotactic sensitivity, a very fine mesh had to be utilized and thus the efficiency of the numerical method was affected. In this work, we consider an alternative approach relying on mesh adaptation, which helps to improve the approximation of the singular structures evolved by chemotaxis models. We develop, in particular, an adaptive moving mesh (AMM) finite-volume semi-discrete upwind method for the two-species chemotaxis system. The proposed AMM technique allows one to increase the density of mesh nodes at the blowup regions. This helps to substantially improve the resolution while using a relatively small number of finite-volume cells.

Published
2019

12. A first-order hyperbolic system approach for dispersion

Author

Alireza Mazaheri, Hiroaki Nishikawa, Mario Ricchiuto, NASA Langley Research Center [Hampton] (LaRC), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), National Institute of Aerospace [Hampton] (NIA), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Hessian matrix, Numerical Analysis, Physics and Astronomy (miscellaneous), Applied Mathematics, Mathematical analysis, Order of accuracy, 010103 numerical & computational mathematics, First order, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 01 natural sciences, Hyperbolic systems, Computer Science Applications, Shock (mechanics), 010101 applied mathematics, Dispersive partial differential equation, Computational Mathematics, symbols.namesake, Modeling and Simulation, symbols, Partial derivative, 0101 mathematics, Dispersion (water waves), ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We propose a new first-order hyperbolic system approach for dispersive partial differential equations.We apply a compact 4th-order RD scheme, and solve time dependent dispersive PDEs.We demonstrate the performance of the high-order RD schemes on the proposed hyperbolic system, including dispersive shock cases.We verify that the predicted solution, its gradient and Hessian have the same order of accuracy on randomly distributed nodes.

Published
2016

13. The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

Author

Alex Main, Mario Ricchiuto, Ting Song, Guglielmo Scovazzi, Duke University [Durham], Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-11-RSNR-0023,TANDEM,Tsunamis en Atlantique et MaNche : Définition des Effets par Modélisation(2011), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Physics and Astronomy (miscellaneous), Discretization, Computation, 010103 numerical & computational mathematics, 01 natural sciences, Robustness (computer science), Boundary value problem, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Shallow water equations, ComputingMilieux_MISCELLANEOUS, [SDU.STU.OC]Sciences of the Universe [physics]/Earth Sciences/Oceanography, Mathematics, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Numerical Analysis, Applied Mathematics, Mathematical analysis, Wave equation, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Waves and shallow water, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We propose a new computational approach for embedded boundary simulations of hyperbolic systems and, in particular, the linear wave equations and the nonlinear shallow water equations. The proposed approach belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. This framework is applied here in the setting of a stabilized finite element method, even though other spatial discretization techniques could have been employed. Accuracy, stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations. Comparisons with standard weak boundary conditions imposed on body-fitted grids, which conform to the geometry of the computational domain boundaries, are also presented.

Published
2018

14. High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Author

Léo Nouveau, Mario Ricchiuto, Guglielmo Scovazzi, Duke University [Durham], Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Finite element method, Diffusion equation, Physics and Astronomy (miscellaneous), Boundary (topology), 010103 numerical & computational mathematics, Computational fluid dynamics, 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], High-order approximation, symbols.namesake, Neumann boundary condition, Applied mathematics, Stabilized methods, Boundary value problem, 0101 mathematics, Variable (mathematics), Mathematics, Numerical Analysis, Applied Mathematics, Darcy flow, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Dirichlet boundary condition, symbols, Heat equation, Embedded boundary, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose an extension of the embedded boundary method known as "shifted boundary method" to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.

Published
2019

15. r−adaptation for Shallow Water flows: conservation, well balancedness, efficiency

Author

Luca Arpaia, Mario Ricchiuto, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Plafrim, CARDAMOM, ANR-11-RSNR-0023,TANDEM,Tsunamis en Atlantique et MaNche : Définition des Effets par Modélisation(2011), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Mathematical optimization, Finite volume method, Partial differential equation, General Computer Science, General Engineering, CPU time, Eulerian path, [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], 010103 numerical & computational mathematics, 01 natural sciences, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, symbols.namesake, Flow (mathematics), symbols, 0101 mathematics, Reduction (mathematics), Shallow water equations, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Interpolation
Abstract

We investigate the potential of the so-called "relocation" mesh adaptation in terms of resolution and efficiency for the simulation of free surface flows in the near shore region. Our work is developed in three main steps. First, we consider several Arbitrary Lagrangian Eulerian (ALE) formulations of the shallow water equations on moving grids, and provide discrete analogs in the Finite Volume and Residual Distribution framework. The compliance to all the physical constraints, often in competition, is taken into account. We consider different formulations allowing to combine volume conservation (DGCL) and equilibrium (Well-Balancedness), and we clarify the relations between the so-called pre-balanced form of the equations (Rogers et al., J.Comput.Phys. 192, 2003), and the classical upwiniding of the bathymetry term gradients (Bermudez and Vazquez, Computers and Fluids 235, 1994). Moreover, we propose a simple remap of the bathymetry based on high accurate quadrature on the moving mesh which, while preserving an accurate representation of the initial data, also allows to retain mass conservation within an arbitrary accuracy. Second, the coupling of the resulting schemes with a mesh partial differential equation is studied. Since the flow solver is based on genuinely explicit time stepping, we investigate the efficiency of three coupling strategies in terms of cost overhead w.r.t. the flow solver. We analyze the role of the solution remap necessary to evaluate the error monitor controlling the adaptation, and propose simplified formulations allowing a reduction in computational cost. The resulting ALE algorithm is compared with the \textit{rezoning} Eulerian approach with interpolation proposed e.g. in (Tang and Tang, SINUM 41, 2003). An alternative cost effective Eulerian approach, still allowing a full decoupling between adaptation and flow evolution steps is also proposed. Finally, a thorough numerical evaluation of the methods discussed is performed. Numerical results on propagation, and inundation problems shows that the best compromise between accuracy and CPU time is provided by a full ALE formulation. If a loose coupling with the mesh adaptation is sought, then the cheaper Eulerian approach proposed is shown to provide results quite close to the ALE.

Published
2018

16. High-Order Methods for CFD

Author

Mario Ricchiuto and Rémi Abgrall

Subjects
Mathematical optimization, 010504 meteorology & atmospheric sciences, business.industry, Numerical analysis, 010103 numerical & computational mathematics, Computational fluid dynamics, Residual, 01 natural sciences, Maximum principle, Flow (mathematics), Path (graph theory), Applied mathematics, 0101 mathematics, business, Focus (optics), Shallow water equations, 0105 earth and related environmental sciences, Mathematics
Abstract

We provide a review of high order methods for CFD. Besides recalling some classical methods, we show a framework allowing on one hand to see and work with these methods under a different light, and on the other to provide a different path to construct numerical methods for flow equations. In particular, we focus on Residual Based techniques, and Residual Distribution methods, as a framework to construct schemes of arbitrary order. The somewhat classical second-order multidimensional upwind fluctuation-splitting/residual-distribution schemes are reviewed in the chapter by Deconinck and Ricchiuto.

Published
2017

17. Shock-fitting and predictor-corrector explicit ALE Residual Distribution

Author

Pierrick Quemar, Lorenzo Campoli, Mario Ricchiuto, Aldo Bonfiglioli, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], EDF R&D (EDF R&D), EDF (EDF), Università degli studi della Basilicata [Potenza] (UNIBAS), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Marcello Onofri, Renato Paciorri, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Predictor–corrector method, Work (thermodynamics), Discretization, High resolution, 01 natural sciences, Residual distribution, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Shock (mechanics), 010101 applied mathematics, Nonlinear system, Applied mathematics, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

This work is part of a systematic evaluation of the unsteady shock-fitting algorithm developed by some of the authors and their co-workers. The objective of the study is to evaluate the impact on the fitted solution of the nature of the discretization method used. In particular, the study considers genuinely multidimensional upwind methods, as well as centred dicretizations with stabilizing terms, and both linear first and second order, and nonlinear high resolution schemes. The methods used belong to the family of Residual Distribution schemes. One- and two-dimensional numerical examples are considered to evaluate the behaviour of the fitted solution both quantitatively and qualitatively.

Published
2017

18. Hyperbolic Method for Dispersive PDEs: Same High-Order of Accuracy for Solution, Gradient, and Hessian

Author

Hiroaki Nishikawa, Alireza Mazaheri, Mario Ricchiuto, NASA Langley Research Center [Hampton] (LaRC), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), National Institute of Aerospace [Hampton] (NIA), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Hessian matrix, FTCS scheme, Hessian equation, Mathematical analysis, First-order partial differential equation, Order of accuracy, 010103 numerical & computational mathematics, 01 natural sciences, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010305 fluids & plasmas, symbols.namesake, Elliptic partial differential equation, Method of characteristics, 0103 physical sciences, symbols, 0101 mathematics, Hyperbolic partial differential equation, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

In this paper, we introduce a new hyperbolic first-order system for general dispersive partial differential equations (PDEs). We then extend the proposed system to general advection-diffusion-dispersion PDEs. We apply the fourth-order RD scheme of Ref. 1 to the proposed hyperbolic system, and solve time-dependent dispersive equations, including the classical two-soliton KdV and a dispersive shock case. We demonstrate that the predicted results, including the gradient and Hessian (second derivative), are in a very good agreement with the exact solutions. We then show that the RD scheme applied to the proposed system accurately captures dispersive shocks without numerical oscillations. We also verify that the solution, gradient and Hessian are predicted with equal order of accuracy.

Published
2016

19. An adaptive, residual based, splitting approach for the penalized Navier Stokes equations

Author

Rémi Abgrall, Cécile Dobrzynski, Mario Ricchiuto, Héloïse Beaugendre, Léo Nouveau, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Polytechnique de Bordeaux (Bordeaux INP), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Universität Zürich [Zürich] = University of Zurich (UZH), PLAFRIM-CPU-MCIA, European Project: 605180,EC:FP7:TPT,FP7-AAT-2013-RTD-1,STORM(2013), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Unstructured grids, Discretization, Computational Mechanics, General Physics and Astronomy, 010103 numerical & computational mathematics, Computational fluid dynamics, Residual, 01 natural sciences, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Boundary value problem, 0101 mathematics, Navier–Stokes equations, ComputingMethodologies_COMPUTERGRAPHICS, Mathematics, Immersed boundary method, business.industry, [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, Mechanical Engineering, Mathematical analysis, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, 010101 applied mathematics, Splitting, Unsteady residual schemes, Strang splitting, Mechanics of Materials, Mesh generation, business, Penalization, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; The interest on Immersed Boundary Methods (IBM) is increasing in Computational Fluid Dynamics as they simplify the mesh generation problem. In this work, we consider an approach based on the addition of a penalty term to the Navier–Stokes equations to account for the wall boundary conditions. To discretize the resulting equations we use a residual distribution approach previously developed by some of the authors. To adapt the method to the IBM considered, we developed a new formulation of residual distribution based on a Strang splitting method in time, coupling an implicit asymptotic integration procedure of the penalization ODE with a simplified explicit residual distribution for the Navier–Stokes equations. The first method, provides an operator which is exact up to orders η2, with η the penalty parameter assuming values of the order of 10−10. A modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes, is also introduced in the paper. This guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid. We show formally and numerically that the approach proposed is second order accurate for smooth solutions. We evaluate its potential for IBM by coupling the resulting method with unstructured mesh adaptation on wall boundaries. Several steady and time dependent tests are used to show the promising features of the method proposed.

Published
2016

20. A second order residual based predictor–corrector approach for time dependent pollutant transport

Author

R. Ata, J.-M. Hervouet, S. Pavan, Mario Ricchiuto, Simulation et Traitement de l'information pour l'Exploitation des systèmes de Production (EDF R&D STEP), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (Saint-Venant), École des Ponts ParisTech (ENPC)-Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema)-EDF R&D (EDF R&D), Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-11-RSNR-0023,TANDEM,Tsunamis en Atlantique et MaNche : Définition des Effets par Modélisation(2011), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Predictor–corrector method, Physics and Astronomy (miscellaneous), [SDU.STU.GP]Sciences of the Universe [physics]/Earth Sciences/Geophysics [physics.geo-ph], Scalar (mathematics), Monotonic function, Residual, 01 natural sciences, 010305 fluids & plasmas, Maximum principle, 0103 physical sciences, Applied mathematics, 0101 mathematics, [SDU.STU.HY]Sciences of the Universe [physics]/Earth Sciences/Hydrology, Mathematics, Numerical Analysis, Finite volume method, Applied Mathematics, Mathematical analysis, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Monotone polygon, Continuity equation, 13. Climate action, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We present a second order residual distribution scheme for scalar transport problems in shallow water flows. The scheme, suitable for the unsteady cases, is obtained adapting to the shallow water context the explicit Runge-Kutta schemes for scalar equations [1]. The resulting scheme is decoupled from the hydrodynamics yet the continuity equation has to be considered in order to respect some important numerical properties at discrete level. Beyond the classical characteristics of the residual formulation presented in [1] and [2], we introduce the possibility to iterate the corrector step in order to improve the accuracy of the scheme. Another novelty is that the scheme is based on a precise monotonicity condition which guarantees the respect of the maximum principle. We thus end up with a scheme which is mass conservative, second order accurate and monotone. These properties are checked in the numerical tests, where the proposed approach is also compared to some finite volume schemes on unstructured grids. The results obtained show the interest in adopting the predictor corrector scheme for pollutant transport applications, where conservation of the mass, monotonicity and accuracy are the most relevant concerns.

Published
2016

21. A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and runup

Author

Andrea Gilberto Filippini, Maria Kazolea, Mario Ricchiuto, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Physics and Astronomy (miscellaneous), Discretization, 01 natural sciences, 010305 fluids & plasmas, 0103 physical sciences, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Dispersion (water waves), ComputingMilieux_MISCELLANEOUS, Mathematics, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Numerical Analysis, Finite volume method, Applied Mathematics, Numerical analysis, Mathematical analysis, Finite difference method, Breaking wave, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Classical mechanics, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper we evaluate hybrid strategies for the solution of the Green–Naghdi system of equations for the simulation of fully nonlinear and weakly dispersive free surface waves. We consider a two step solution procedure composed of: a first step where the non-hydrostatic source term is recovered by inverting the elliptic coercive operator associated to the dispersive effects; a second step which involves the solution of the hyperbolic shallow water system with the source term, computed in the previous phase, which accounts for the non-hydrostatic effects. Appropriate numerical methods, that can be also generalized on arbitrary unstructured meshes, are used to discretize the two stages: the standard C 0 Galerkin finite element method for the elliptic phase; either third order Finite Volume or third order stabilized Finite Element method for the hyperbolic phase. The discrete dispersion properties of the fully coupled schemes obtained are studied, showing accuracy close to or better than that of a fourth order finite difference method. The hybrid approach of locally reverting to the nonlinear shallow water equations is used to recover energy dissipation in breaking regions. To this scope we evaluate two strategies: simply neglecting the non-hydrostatic contribution in the hyperbolic phase; imposing a tighter coupling of the two phases, with a wave breaking indicator embedded in the elliptic phase to smoothly turn off the dispersive effects. The discrete models obtained are thoroughly tested on benchmarks involving wave dispersion, breaking and run-up, showing a very promising potential for the simulation of complex near shore wave physics in terms of accuracy and robustness.

Published
2016

22. Numerical approximation of parabolic problems by residual distribution schemes

Author

Rémi Abgrall, G Baurin, Arnaud Krust, Mario Ricchiuto, and Dante De Santis

Subjects
Compact stencil, Applied Mathematics, Mechanical Engineering, Mathematical analysis, Scalar (mathematics), Computational Mechanics, 010103 numerical & computational mathematics, Classification of discontinuities, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, Nonlinear system, Numerical approximation, Mechanics of Materials, Inviscid flow, Computational mechanics, Polygon mesh, 0101 mathematics, Mathematics
Abstract

SUMMARY We are interested in the numerical approximation of steady scalar convection–diffusion problems by means of high order schemes called Residual Distribution schemes. In the inviscid case, one can develop nonlinear Residual Distribution schemes that are nonoscillatory, even in the case of very strong discontinuities, while having the most possible compact stencil, on hybrid unstructured meshes. This paper proposes and compare extensions of these schemes for the convection–diffusion problem. This methodology, in particular in terms of accuracy, is evaluated on problem with exact solutions. Its nonoscillatory behavior is tested against the Smith and Hutton problem. Copyright © 2012 John Wiley & Sons, Ltd.

Published
2012

23. Third order residual distribution schemes for the Navier–Stokes equations

Author

N. Villedieu, Tiago Quintino, Mario Ricchiuto, Herman Deconinck, von Karman Institute for Fluid Dynamics (VKI), Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics and Astronomy (miscellaneous), Discretization, Boundary (topology), Upwind scheme, Residual, 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Mathematics::Numerical Analysis, 010305 fluids & plasmas, Physics::Fluid Dynamics, symbols.namesake, Inviscid flow, 0103 physical sciences, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Navier–Stokes equations, ComputingMilieux_MISCELLANEOUS, Mathematics, Numerical Analysis, Applied Mathematics, Mathematical analysis, 1. No poverty, Reynolds number, Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P2 Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number.

Published
2011

24. Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes

Author

Rémi Abgrall, Adam Larat, Mario Ricchiuto, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Farhat Research Group [Stanford] (FRG), Stanford University, Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Work (thermodynamics), Mathematical optimization, Physics and Astronomy (miscellaneous), Generalization, 010103 numerical & computational mathematics, 01 natural sciences, Residual distribution, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], symbols.namesake, Inviscid flow, Applied mathematics, Polygon mesh, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Mathematics, Numerical Analysis, Applied Mathematics, Finite element method, Computer Science Applications, Euler equations, 010101 applied mathematics, Computational Mathematics, Third order, Modeling and Simulation, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the residual distribution method of (Abgrall, J.ComputPhys 2006) to very high order of accuracy, by extending the preliminary work discussed in (Abgrall, Larat, Ricchiuto, Tave, Computers and Fluids 2009) to systems and hybrid meshes. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we both have a non-oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems.

Published
2011

25. Explicit Runge–Kutta residual distribution schemes for time dependent problems: Second order case

Author

Mario Ricchiuto and Rémi Abgrall

Subjects
Numerical Analysis, Finite volume method, Physics and Astronomy (miscellaneous), Discretization, Applied Mathematics, Mathematical analysis, Context (language use), 010103 numerical & computational mathematics, Residual, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Runge–Kutta methods, Operator (computer programming), Modeling and Simulation, Probability distribution, 0101 mathematics, Galerkin method, Mathematics
Abstract

In this paper, we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems, starting from a given residual distribution (RD) discrete approximation of the steady operator. We review the existing knowledge on consistent RD mass matrices and highlight the relations between different definitions. We then introduce our explicit approach which is based on three main ingredients: first recast the RD discretization as a stabilized Galerkin scheme, then use a shifted time discretization in the stabilization operator, and lastly apply high order mass lumping on the Galerkin component of the discretization. The discussion is particularly relevant for schemes of the residual distribution type [18,3] which we will use for all our numerical experiments. However, similar ideas can be used in the context of residual-based finite volume discretizations such as the ones proposed in [14,12]. The schemes are tested on a wide variety of classical problems confirming the theoretical expectations.

Published
2010

26. Stabilized residual distribution for shallow water simulations

Author

Andreas Bollermann, Mario Ricchiuto, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut für Geometrie und Praktische Mathematik [RWTH Aachen] (IGPM), Rheinisch-Westfälische Technische Hochschule Aachen, Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH)

Subjects
Numerical Analysis, Physics and Astronomy (miscellaneous), Discretization, Applied Mathematics, Mathematical analysis, 010103 numerical & computational mathematics, Type (model theory), Residual, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 01 natural sciences, Residual distribution, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Waves and shallow water, Operator (computer programming), Modeling and Simulation, 0101 mathematics, Shallow water equations, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We propose a stabilized Residual Distribution (RD) scheme for the simulation of shallow water flows. The final discretization is obtained combining the stabilized RD approach proposed in (Abgrall, J. Comp. Phys. 214, 2006) and (Ricchiuto and Abgrall, ICCFD4, Springer-Verlag 2006), with the conservative formulation already used in (Ricchiuto et al., J. Comp. Phys. 222, 2007) to simulate shallow water flows. The scheme proposed is a nonlinear variant of a Lax-Friedrichs type discretization. It is well balanced, it actually yields second-order of accuracy in smooth areas, and it preserves the positivity of the height of the water in presence of dry areas. This is made possible by the residual character of the discretization, by properly adapting the stabilization operators proposed in (Abgrall, J. Comp. Phys. 214, 2006) and (Ricchiuto and Abgrall, ICCFD4, Springer-Verlag, 2006), and thanks to the positivity preserving character of the underlying Lax-Friedrichs scheme. We demonstrate the properties of the discretization proposed on a wide variety of tests.

Published
2009

27. On uniformly high-order accurate residual distribution schemes for advection–diffusion

Author

N. Villedieu, Mario Ricchiuto, Herman Deconinck, Rémi Abgrall, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Algorithms and high performance computing for grand challenge applications (SCALAPPLIX), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), von Karman Institute for Fluid Dynamics (VKI), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Inria Bordeaux - Sud-Ouest, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Sciences et Technologies - Bordeaux 1-Université Bordeaux Segalen - Bordeaux 2

Subjects
residual distribution, computer.software_genre, Topology, 01 natural sciences, 010305 fluids & plasmas, 0103 physical sciences, Advection–diffusion, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, 0101 mathematics, Scaling, Condition number, Mathematics, numerical examples, Numerical linear algebra, Compact stencil, Numerical analysis, Applied Mathematics, Finite element method, 010101 applied mathematics, steady advection-diffusion, Computational Mathematics, Distribution function, finite elements, very high-order schemes, compact schemes, upwind stabilization, Viscosity solution, computer
Abstract

International audience; We discuss preliminary results on the construction of uniformly high-order residual distribution $(\cal RD)$ type discretizations for steady advection-diffusion on unstructured grids. A properly designed scaling of the $(\cal RD)$ upwind stabilization with the physical viscosity allows to obtain schemes with uniform and arbitrary accuracy, on a very compact stencil. Second- and third-order examples are given to illustrate the potential of the approach.

Published
2008
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28. High-order residual distribution and error estimation for steady and unsteady compressible flow

Author

N. Villedieu, Mario Ricchiuto, Lila Kolozar, Stefano D'Angelo, Martin Vymazal, Herman Deconinck, Aeronautics and Aerospace Department [Rhode-St-Genèse], von Karman Institute for Fluid Dynamics (VKI), CERFACS, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique (CERFACS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Distribution (number theory), Mathematical analysis, 010103 numerical & computational mathematics, 01 natural sciences, Compressible flow, Residual distribution, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010101 applied mathematics, [SPI]Engineering Sciences [physics], Acoustic wave propagation, Test case, 0101 mathematics, High order, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

In the first part, an extension of upwind residual distribution schemes for high-order accurate solution of hyperbolic problems is introduced, based on the use of spatially varying distribution matrices. Following this, the application to adjoint-based error estimation for steady compressible flow is presented. Finally the resolution of acoustic wave propagation by a space-time residual distribution is discussed. The accuracy of the methodology is demonstrated on several test cases.

Published
2015

29. Runge–Kutta Residual Distribution Schemes

Author

Matthew E. Hubbard, Andrzej Warzyński, Mario Ricchiuto, School of Computing [Leeds], University of Leeds, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discretization, 010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, Theoretical Computer Science, Applied mathematics, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Numerical Analysis, Partial differential equation, Applied Mathematics, Mathematical analysis, Linear system, General Engineering, Mass matrix, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Computational Mathematics, Runge–Kutta methods, Character (mathematics), Computational Theory and Mathematics, Hyperbolic conservation laws, Time-dependent problems, Second order schemes, Residual distribution, Runge–Kutta time-stepping, Variety (universal algebra), Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge---Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

Published
2015

30. Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries

Author

Andrea Gilberto Filippini, Mario Ricchiuto, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics and Astronomy (miscellaneous), Discretization, Wave propagation, 010103 numerical & computational mathematics, Residual, 01 natural sciences, Residual distribution, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Nonlinear shallow water equations, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Boussinesq approximation (water waves), Shallow water equations, ComputingMilieux_MISCELLANEOUS, Mathematics, Numerical Analysis, [SDE.IE]Environmental Sciences/Environmental Engineering, Applied Mathematics, Mathematical analysis, Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, [SPI.GCIV]Engineering Sciences [physics]/Civil Engineering, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this paper we consider the solution of the enhanced Boussinesq equations of Madsen and Sorensen (1992) [55] by means of residual based discretizations. In particular, we investigate the applicability of upwind and stabilized variants of continuous Galerkin finite element and Residual Distribution schemes for the simulation of wave propagation and transformation over complex bathymetries. These techniques have been successfully applied to the solution of the nonlinear Shallow Water equations (see e.g. Hauke (1998) [39] and Ricchiuto and Bollermann (2009) [61]). In a first step toward the construction of a hybrid model coupling the enhanced Boussinesq equations with the Shallow Water equations in breaking regions, this paper shows that equal order and even low order (second) upwind/stabilized techniques can be used to model non-hydrostatic wave propagation over complex bathymetries. This result is supported by theoretical (truncation and dispersion) error analyses, and by thorough numerical validation.

Published
2014

31. High Order Nonlinear Numerical Schemes for Evolutionary PDEs

Author

Rémi Abgrall, Héloïse Beaugendre, Cécile Dobrzynski, Mario Ricchiuto, Pietro Marco Congedo, and Vincent Perrier

Subjects
010101 applied mathematics, Nonlinear system, Computer science, 0103 physical sciences, Applied mathematics, 0101 mathematics, High order, 01 natural sciences, 010305 fluids & plasmas
Published
2014

32. Comparison of high order algorithms in Aerosol and Aghora for compressible flows

Author

F. Renac, Maxime Mogé, Mario Ricchiuto, Emeric Martin, Francois Rué, Cedric Lachat, D. A. Mbengoue, Vincent Perrier, Damien Genet, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Control, Analysis and Simulations for TOkamak Research (CASTOR), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), ONERA - The French Aerospace Lab [Châtillon], ONERA-Université Paris Saclay (COmUE), Laboratoire de Mathématiques appliquées de Pau (LMAP), Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA), Computational Approximation with discontinous Galerkin methods and compaRison with Experiments (CAGIRE), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Service Expérimentation et Développement [Bordeaux] (SED), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
T57-57.97, Applied mathematics. Quantitative methods, Operations research, Computer science, 010103 numerical & computational mathematics, 01 natural sciences, Finite element method, Computational science, Aerosol, 010101 applied mathematics, QA1-939, Compressibility, 0101 mathematics, High order, Implementation, Mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This article summarizes the work done within the Colargol project during CEMRACS 2012. The aim of this project is to compare the implementations of high order finite element methods for compressible flows that have been developed at ONERA and at INRIA for about one year, within the Aghora and Aerosol libraries.

Published
2013

33. An ALE formulation for explicit Runge-Kutta Residual Distribution

Author

Remi Abgrall, Luca Arpaia, Mario Ricchiuto, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), INRIA, Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010101 applied mathematics, [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing), [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 010103 numerical & computational mathematics, 0101 mathematics, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper we consider the solution of hyperbolic conservation laws on moving meshes by means of an Arbitrary Lagrangian Eulerian (ALE) formulation. In particular we propose an ALE framework for the genuinely explicit residual distribution schemes of (Ricchiuto and Abgrall {\it J.Comput.Phys} 229, 2010). The discretizations obtained are thoroughly tested on a large number of benchmarks.; Dans ce travail on considére la resolution de lois de conservation sur maillages mobiles par une formulation Arbitrary Lagrangian Eulerian (ALE). On propose en particulier un formalisme ALE pour les schémas RD explicites de (Ricchiuto and Abgrall {\it J.Comput.Phys} 229, 2010). Les schémas ainsi obtenus sont testés sur des nombreux benchmarks.

Published
2013

34. Discontinuous upwind residual distribution: A route to unconditional positivity and high order accuracy

Author

Matthew E. Hubbard, Mario Ricchiuto, School of Computing [Leeds], University of Leeds, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Conservation law, General Computer Science, Spacetime, business.industry, Mathematical analysis, Scalar (mathematics), General Engineering, 010103 numerical & computational mathematics, Computational fluid dynamics, 01 natural sciences, Finite element method, Burgers' equation, Euler equations, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010101 applied mathematics, symbols.namesake, Mesh generation, symbols, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, business, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, ComputingMethodologies_COMPUTERGRAPHICS
Abstract

International audience; This paper proposes an approach to the approximation of time-dependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous-in-time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers' equation and the Euler equations of gas dynamics.

Published
2011

35. Some examples of high order simulations parallel of inviscid flows on unstructured and hybrid meshes by residual distribution schemes

Author

Guillaume Baurin, Mario Ricchiuto, Pascal Jacq, Rémi Abgrall, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), SNECMA Villaroche [Moissy-Cramayel], Safran Group, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Inria Bordeaux - Sud-Ouest, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Computer Science, General Engineering, 010103 numerical & computational mathematics, Volume mesh, Topology, 01 natural sciences, Computational science, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010101 applied mathematics, Third order, symbols.namesake, Inviscid flow, Euler's formula, symbols, Code (cryptography), Development (differential geometry), Polygon mesh, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Block (data storage), ComputingMethodologies_COMPUTERGRAPHICS
Abstract

International audience; Our aim is to report some recent advances in the development of residual distribution (RD) schemes using unstructured meshes: we present here some 3D results using pure tet meshes with a third order accurate scheme and 3D results using meshes with hex only. These latter meshes originate from ONERA where they have been used for Euler simulations with the Elsa code. Elsa only uses block structured meshes so that we have transformed the "ijk" format of the mesh into a nonstructured one without modifying the location of vertices and the connectivity of the mesh, so that it is exactly the same mesh.

Published
2011

36. On the C-property and generalized C-property of Residual Distribution for the Shallow Water equations

Author

Mario Ricchiuto, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Discretization, Applied Mathematics, Mathematical analysis, General Engineering, Residual, 01 natural sciences, Residual distribution, 010305 fluids & plasmas, Theoretical Computer Science, Quadrature (mathematics), 010101 applied mathematics, Computational Mathematics, Waves and shallow water, Computational Theory and Mathematics, 0103 physical sciences, 0101 mathematics, Shallow water equations, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; In this paper we consider the discretization the Shallow Water equa- tions by means of Residual Distribution (RD) schemes, and review the condi- tions allowing the exact preservation of some exact steady solutions. These conditions are shown to be related to both the type of spatial approximation and to the quadrature used to evaluate the cell residual. Numerical examples are shown to validate the theory.

Published
2010

37. Implicit Strategy and Parallelization of a High Order Residual Distribution Scheme

Author

Adam Larat, Pascal Jacq, Rémi Abgrall, Xavier Lacoste, Rémi Butel, Mario Ricchiuto, Cedric Lachat, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Plasma, tUrbulence, Modeling, Approximation and Simulation (PUMAS), Inria Sophia Antipolis - Méditerranée (CRISAM), Farhat Research Group [Stanford] (FRG), Stanford University, Kroll, N., Bieler, H., Deconinck, Couaillier, V., Ven, H. van der, Sorensen, K., European Project: 36873,ADIGMA, Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Scheme (programming language), Order (ring theory), 010103 numerical & computational mathematics, Parallel computing, 01 natural sciences, Residual distribution, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Automatic parallelization, Linear algebra, 0101 mathematics, computer, Mathematics, computer.programming_language
Abstract

This chapter describes the implicitation and parallelization strategies we have developed, as well as some development and results using high performance linear algebra softwares.

Published
2010

38. Construction of High-Order Non Upwind Distribution Schemes

Author

Mario Ricchiuto, Adam Larat, and Rémi Abgrall

Subjects
Generalization, business.industry, Upwind scheme, Laminar flow, 010103 numerical & computational mathematics, Computational fluid dynamics, 01 natural sciences, Finite element method, Euler equations, 010101 applied mathematics, symbols.namesake, Distribution (mathematics), Mach number, symbols, Applied mathematics, 0101 mathematics, business, Mathematics
Abstract

In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the Residual Distribution method of [8] to very high order of accuracy, by extending the preliminary work discussed in [17]. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we an both have a non oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems. We also extend the scheme to laminar viscous problems.

Published
2010

39. A simple construction of very high order non oscillatory compact schemes on unstructured meshes

Author

Mario Ricchiuto, Adam Larat, Rémi Abgrall, Cédric Tavé, Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems (BACCHUS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), ADIGMA, Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Computer Science, Scalar (mathematics), General Engineering, ACM, Topology, 01 natural sciences, Residual distribution, 010305 fluids & plasmas, 010101 applied mathematics, 0103 physical sciences, Applied mathematics, Polygon mesh, 0101 mathematics, High order, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

In [Abgrall R, Roe PL. High order fluctuation schemes on triangular meshes. J Sci Comput 2003;19(1–3):3–36] have been constructed very high order residual distribution schemes for scalar problems. They were using triangle unstructured meshes. However, the construction was quite involved and was not very flexible. Here, following [Abgrall R. Essentially non-oscillatory residual distribution schemes for hyperbolic problems. J Comput Phys 2006;214(2):773–808], we develop a systematic way of constructing very high order non-oscillatory schemes for such meshes. Applications to scalar and systems problems are given.

Published
2009

40. Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes

Author

Rémi Abgrall, Herman Deconinck, Mario Ricchiuto, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Algorithms and high performance computing for grand challenge applications (SCALAPPLIX), INRIA Futurs, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), von Karman Institute for Fluid Dynamics (VKI), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Conservation law, Mathematical optimization, Physics and Astronomy (miscellaneous), Discretization, Applied Mathematics, Mesh networking, Linearity, 010103 numerical & computational mathematics, 01 natural sciences, Residual distribution, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, Polynomial interpolation, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Applied mathematics, Polygon mesh, 0101 mathematics, Shallow water equations, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We consider the numerical solution of the shallow water equations on unstructured grids. We focus on flows over wet areas. The extension to the case of dry bed will be reported elsewhere. The shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding, in this case, with the total energy. Our aim is to show the application of a particular class of conservative residual distribution (RD) schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties. We give a review of conservative RD schemes showing relations between different approaches previously published, and recall L^~ stability and accuracy criteria characterizing the schemes. In particular, the accuracy of the RD method in presence of source terms is analyzed, and conditions to construct rth order discretizations on irregular triangular grids are proved. It is shown that the RD approach gives a natural way of obtaining high order discretizations which, moreover, preserves exactly the steady lake at rest solution independently on mesh topology, nature of the variation of the bottom and polynomial order of interpolation used for the unknowns. We also consider more general analytical solutions which are less investigated from the numerical view point. On irregular grids, linearity preserving RD schemes yield a truly second order approximation. We also sketch a strategy to achieve discretizations which preserve exactly some of these solutions. Numerical results on steady and time-dependent problems involving smooth and non-smooth variations of the bottom topology show very promising features of the approach.

Published
2007

41. Residual Distribution Schemes: Foundations and Analysis

Author

Herman Deconinck, Mario Ricchiuto, von Karman Institute for Fluid Dynamics (VKI), Algorithms and high performance computing for grand challenge applications (SCALAPPLIX), INRIA Futurs, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematical optimization, Finite volume method, Discretization, Truncation error (numerical integration), Numerical analysis, Upwind scheme, 010103 numerical & computational mathematics, 01 natural sciences, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Finite element method, 010305 fluids & plasmas, Euler equations, 010101 applied mathematics, symbols.namesake, Maximum principle, 0103 physical sciences, symbols, Applied mathematics, 0101 mathematics, Mathematics
Abstract

In this article, we introduce and analyze the class of numerical schemes known as residual distribution or fluctuation splitting schemes. The root of this family of discretizations is found mainly in works aiming at generalizing the finite volume techniques to a genuinely multidimensional upwinding context, as in, for example (Hall, Morton, Ni, Roe). However, the final result is a numerical method sharing a lot with other techniques, such as finite element schemes (Carette, Deconinck, Paillere and Roe, 1995). Our aim is to look at fluctuation splitting/residual distribution methods from a generic theoretical point of view. We discuss in a general way the basic principles and the criteria used in the design of the schemes: positivity, k-exacteness, energy stability, well-balancedness. We also show similarities with known, more classical, discretization techniques, as well as the main distinguishing features that make residual distribution methods a valid alternative to these classical schemes. Finally, we present some numerical applications and comparisons. Throughout the text, we give an extensive overview of the existing literature, and finally conclude the chapter by reviewing most of the ongoing research on the topic. Keywords: residual distribution; fluctuation splitting; unstructured meshes; high order of accuracy; k-exactness; truncation error analysis; positivity; discrete maximum principle; energy stability; multidimensional upwinding; conservation; space-time schemes; matrix schemes; Euler equations; homogeneous two-phase model; shallow water equations

Published
2007

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