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We present a new multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations. This scheme relies on relaxation and splitting techniques and can be easily used at high order. A fully conservative version is not entropy satisfying but is observed experimentally to be more robust than standard constrained transport schemes at low plasma beta. At very low plasma beta and high Alfvén number, we have designed an entropy-satisfying version that is not conservative for the magnetic field but preserves admissible states and we switch locally a-priori between the two versions depending on the regime of plasma beta and Alfvén number. This strategy is robust in a wide range of standard MHD test cases, all performed at second order with a classic MUSCL-Hancock scheme. • This paper presents a new multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations. • The role of the zero-divergence of the magnetic field is re-investigated with an entropy-satisfying numerical scheme. • This scheme relies on modern relaxation and splitting techniques and can be easily used at high order. [ABSTRACT FROM AUTHOR]

This paper describes a novel approach to alleviate the carbuncle phenomenon which consists in adding to any carbuncle prone Riemann solver an extra viscosity term in tangential momentum flux and its contribution to the energy conservation equation. This term contains one numerical parameter only, a scalar viscosity, which is reduced using a face-based shear detector to preserve shear waves. The idea stems from the investigation of some of the existing Riemann solvers, also presented in the paper. Indeed, when splitting the numerical flux into the face normal and tangential components, we observe that all the carbuncle free Riemann solvers present in the tangential part a numerical viscosity which scales with the sound speed when the normal flow velocity becomes zero. Opposite, in the carbuncle prone solvers this viscosity scales with the normal flow velocity. In particular the carbuncle free HLLCM scheme proposed by Shen et al. can be written by adding to the carbuncle prone HLLC scheme a tangential artificial viscosity term. Then the same can be done for any other Riemann solver, which renders the approach easy to implement in CFD codes for compressible flows. Numerical experiments shows the efficiency of the approach in computing carbuncle free single-component and multi-material flows. • Some Riemann solvers are investigated by splitting their numerical flux into interface normal and tangential components. • The carbuncle free ones present a tangential viscosity which scales with the sound speed as the normal velocity vanishes. • A tangential artificial viscosity approach is then proposed to alleviate the carbuncle problem to any Riemann solver. • The approach, combined with a shear sensor to preserve shear waves, is easy to implement in CFD codes. [ABSTRACT FROM AUTHOR]

In this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM) models in fractured porous media accounting for Coulomb frictional contact at matrix fracture interfaces. The simulation of such models plays an important role in many applications such as hydraulic stimulation in deep geothermal systems and assessing induced seismic risks in CO 2 storage. We first extend to the mixed-dimensional framework the thermodynamically consistent THM models derived in [19] based on first and second principles of thermodynamics. Two formulations of the energy equation will be considered based either on energy conservation or on the entropy balance, assuming a vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations preserving energy estimates for both types of formulations and for a general single phase fluid thermodynamical model. This is achieved by a Finite Volume discretisation of the non-isothermal flow based on coercive fluxes and a tailored discretisation of the non-conservative convective terms. It is combined with a mixed Finite Element formulation of the contact-mechanical model with face-wise constant Lagrange multipliers accounting for the surface tractions, which preserves the dissipative properties of the contact terms. The discretisations of both THM formulations are investigated and compared in terms of convergence, accuracy and robustness on 2D test cases. It includes a Discrete Fracture Matrix model with a convection dominated thermal regime, and either a weakly compressible liquid or a highly compressible gas thermodynamical model. • Derivation of mixed-dimensional THM models with frictional contact. • Exploration of models based on energy conservation or entropy balance formulations. • Finite Element - Finite Volume discretisations preserving energy estimates. • Numerical comparison of models for both weakly and highly compressible fluids. [ABSTRACT FROM AUTHOR]

This article provides the high-order finite difference and finite volume ghost multi-resolution weighted essentially non-oscillatory (GMR-WENO) schemes for solving hyperbolic conservation laws on structured meshes. We only utilize the information defined on one big central spatial stencil without introducing any equivalent multi-resolution representations. These GMR-WENO schemes utilize orthogonal Legendre basis to construct high degree polynomials of big central spatial stencils and use a L 2 projection methodology to obtain a series of ghost low degree polynomials whose degree could gradually range from the highest degree to the zeroth degree. Each of these GMR-WENO schemes only utilizes one big central spatial stencil to achieve arbitrary high-order accuracy in smooth regions and could gradually reduce to the first-order accuracy nearby strong discontinuities. The linear weights of such GMR-WENO schemes can be any positive numbers with a summation of one. This is the first time that only one central spatial stencil is utilized in designing high-order finite difference and finite volume WENO schemes. These GMR-WENO schemes have a simple construction and can easily achieve arbitrary high-order accuracy in higher dimensions. Benchmark examples are provided to demonstrate the efficiency of these GMR-WENO schemes. [ABSTRACT FROM AUTHOR]

Efficient modeling of flow physics is a prerequisite for a reliable computation of free-surface environmental flows. Non-hydrostatic flows are often present in shallow water environments, making the task challenging. In this work, we use the method of weighted residuals for modeling non-hydrostatic free surface flows in a depth-averaged framework. In particular, we focus on the Vertically Averaged and Moment (VAM) equations model. First, a new derivation of the model is presented using expansions of the field variables in sigma-coordinates with Legendre polynomials basis. Second, an efficient two-step numerical scheme is proposed: the first step corresponds to solving the hyperbolic part with a second-order path-conservative PVM scheme. Then, in a second step, non-hydrostatic terms are corrected by solving a linear Poisson-like system using an iterative method, thereby resulting in an accurate and efficient algorithm. The computational effort is similar to the one required for the well-known Serre-Green-Naghdi (SGN) system, while the results are largely improved. Finally, the physical aspects of the model are compared to the SGN system and a multilayer model, demonstrating that VAM is comparable in physical accuracy to a two-layer model. • New reformulation of the Vertically Averaged and Moment (VAM) equations model. • Discretization with high order accurate finite. • volume finite-difference scheme. • The proposed approach is computationally very efficient and robust. • Successful comparison with analytical solutions and test problem with experimental reference data. [ABSTRACT FROM AUTHOR]

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms, accommodating the multi-scale nature of the governing equations, which involve both time scales of diffusion and advection operators. The finite volume approach for both explicit and implicit terms allows to encode into the nonlinear flux the velocity of displacement of the Chimera mesh via integration on moving cells. The numerical solution is then projected onto the physically meaningful solution manifold of non-solenoidal fields that stems from the energy equation. To attain second-order time accuracy, we employ semi-implicit IMEX Runge-Kutta schemes. These novel schemes are combined with a fractional-step method, thus the governing equations are eventually solved using a projection method to satisfy the divergence-free constraint of the velocity field. The implicit discretization of the viscous terms allows the CFL-type stability condition for the maximum admissible time step to be only defined by the relative fluid velocity referred to the movement of the frame and not depending on the viscous terms. Communication between different grid blocks is enabled through compact exchange of information from the fringe cells of one mesh block to the field cells of the other block. The continuity of the solution is recovered in one-shot during the solution of the arising algebraic systems by not involving neither direct discretization of the differential operators on fringe cells nor an iterative Schwartz-type method. Free-stream preservation property, i.e. compliance with the Geometric Conservation Law (GCL), is respected at the order of the scheme. The accuracy and capabilities of the new numerical schemes are proved through an extensive range of test cases, demonstrating ability to solve relevant benchmarks in the field of incompressible fluids. • Arbitrary-Lagrangian-Eulerian solver on Chimera meshes. • Compact exchange of information between different mesh blocks. • Implicit-Explicit discretization for multi-scale PDE. • Asymptotically Accurate finite volume schemes. • Incompressible Navier-Stokes system. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a framework in which one can identify and predict the numerical instability of the steady state solution due to the solution reconstruction for cell-centered finite volume methods on unstructured meshes and to stabilize the problem by optimizing the reconstruction stencil. In this work, we first develop and extend a mathematical method, introduced by Haider and his colleagues, to measure the stability impact of the reconstruction phase for both linear and nonlinear problems regardless of the solution. Second order and third order accurate advection and Burgers problems as well as second order Euler problems are used to present detailed practical results and discussion around the use of the local reconstruction map for stability analysis. This method shows that for a range of different physical problems, increasing the stencil size will usually lead to more stable problems. Additionally, an empirical study is performed which sheds light on connections between the mesh properties and the stability of the reconstruction, which in turn helps choose the reconstruction stencil more wisely. Secondly, we propose a systematic approach to optimize both the shape and the size of the reconstruction stencil for better numerical stability through eigenvalue analysis. In this approach, one can directly optimize the solution reconstruction stencil for every control volume to obtain better numerical stability and convergence properties for steady state problems. A second order accurate Euler problem as well as a third order accurate laminar Navier-Stokes problem are used to showcase the applicability of the algorithm. • Identifying and predicting the numerical stability due to the solution reconstruction. • Better numerical stability through eigenvalue analysis and optimization of the solution reconstruction. • Extending solution reconstruction map theory to measure the impact of reconstruction for nonlinear and higher order problems. [ABSTRACT FROM AUTHOR]

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed "multi-resolution WENO schemes", we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes. • A new type of multi-resolution WENO schemes on triangular meshes has been developed. • High order WENO schemes using unequal-sized central spatial stencils have been designed. • The main advantages are compactness, robustness, and good steady-state convergence. [ABSTRACT FROM AUTHOR]

Abstract In this paper we develop explicit local time-stepping (LTS) schemes with second and third order accuracy for the shallow water equations. The system is discretized in space by a C-grid staggering method, namely the TRiSK scheme adopted in MPAS-Ocean, a global ocean model with the capability of resolving multiple resolutions within a single simulation. The time integration is designed based on the strong stability preserving Runge–Kutta (SSP-RK) methods, but different time step sizes can be used in different regions of the domain through the coupling of coarse-fine time discretizations on the interface, and are only restricted by respective local CFL conditions. The proposed LTS schemes are of predictor–corrector type in which the predictors are constructed based on Taylor series expansions and SSP-RK stepping algorithms. The schemes preserve some important physical quantities in the discrete sense, such as exact conservation of the mass and potential vorticity and conservation of the total energy within time truncation errors. Moreover, they inherit the natural parallelism of the original explicit global time-stepping schemes. Extensive numerical tests are presented to illustrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Abstract A space–time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes, endowed with cell average multiresolution analysis for triggering the dynamical grid adaptation. The explicit time scheme features a natural extension of Runge–Kutta methods which allow local time-stepping while guaranteeing accuracy. The use of a compact Runge–Kutta formulation permits further memory reduction. The precision and computational efficiency of the scheme regarding CPU time and memory compression are assessed for problems in one, two and three space dimensions. As application Burgers equation, reaction–diffusion equations and the compressible Euler equations are considered. The numerical results illustrate the efficiency and superiority of the proposed local time-stepping method with respect to the reference computations. Highlights • An efficient local time-stepping approach using Runge–Kutta methods is presented. • The CPU time is reduced in comparison with similar space and time adaptive methods. • The approach maintains similar approximation errors in relation with other methods. • Great performance is obtained for several test problems in 2 and 3 dimensions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a novel mapping approach through the use of adaptive mapping functions is introduced for high order weighted essentially non-oscillatory (WENO) methods. The new class of adaptive mapping functions are designed to adjust themselves to the solution based on a simple parameter calculated using the smoothness indicators that are readily available during computation. It is shown that this adaptive nature allows the resultant mapped WENO scheme to maintain sub-stencil weights close to the optimal weights in smooth regions without amplifying the weights of non-smooth stencils containing discontinuities. Therefore, adaptive mapping achieves enhanced accuracy in smooth regions and is more resistant against spurious oscillations near discontinuities. Taylor series analysis of the seventh order finite volume WENO scheme has been performed to demonstrate the loss of accuracy of the original WENO method near critical points. The convergence rates of the seventh order finite volume WENO scheme with adaptive mapping have been shown through a simple numerical example. Excellent results have been obtained for one-dimensional linear advection cases especially over long output times. Improved results have also been obtained for one- and two-dimensional Euler equation test cases. Highlights • Derivation and comprehensive Taylor series analysis of smoothness indicators of seventh order WENO scheme. • Assessment of convergence properties of finite volume schemes at critical points using a novel, systematic approach. • A generic easy-to-implement class of adaptive mapping functions. [ABSTRACT FROM AUTHOR]

Highlights • High-order accuracy in space is done via Taylor series, whereas in time it is reached through Runge–Kutta methods. • Weak consistency of staggered schemes is proved for barotropic hydrodynamics. • Weak consistency is proved for compressible hydrodynamics using an a posteriori internal energy corrector. • Wide variety of numerical test-cases illustrate the interest, the robustness and the convergence of the schemes. Abstract Staggered grids schemes, formulated in internal energy, are commonly used for CFD applications in industrial context. Here, we prove the consistency of a class of high-order Lagrange-Remap staggered schemes for solving the Euler equations in 1D and 2D on Cartesian grids. The main result of the paper is that using an a posteriori internal energy corrector, the Lagrangian schemes are proved to be conservative in mass, momentum and total energy and to be weakly consistent with the 1D Lagrangian formulation of the Euler equations. Extension in 2D is done using directional splitting methods and face-staggering. Numerical examples in both 1D and 2D illustrate the accuracy, the convergence and the robustness of the schemes. [ABSTRACT FROM AUTHOR]

We extend the unstructured LE vel set / fro NT tracking (LENT) method [1,2] for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4–8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed ρ LENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ρ − / ρ + ∈ [ 1 , 10 4 ]. Numerical stability is demonstrated for in terms of the relative L ∞ velocity error norm, for density-ratios in the range of [ 1 , 10 4 ] , dynamic viscosity-ratios in the range of [ 1 , 10 4 ] and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed ρ LENT method1 is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm. • Exact numerical consistency recovered for the two-phase momentum convection. • Theoretical basis provided for the use of the auxiliary mass conservation equation. • Provided a strict relationship between the mass flux density and the phase indicator. • Implicit discretization of the two-phase momentum convective term. • Increased numerical stability with large surface tension forces and density-ratios. [ABSTRACT FROM AUTHOR]

In this short note, we discuss the circumstances that can lead to a failure to observe the design order of discretization error convergence in accuracy verification when solving a time-dependent problem. In particular, we discuss the problem of failing to observe the design order of spatial accuracy with an extremely small time step. The same problem is encountered even if the time step is reduced with grid refinement. These can cause a serious problem because then one would wind up trying to find a coding error that does not exist. This short note clarifies the mechanism causing this failure and provides a guide for avoiding such pitfalls. • Accuracy verification can easily fail for time-dependent problem. • Counterintuitively, a smaller time step leads to one-order lower accuracy. • The mechanism behind the failure is revealed. • It leads to a practical guid for avoiding the pitfall. [ABSTRACT FROM AUTHOR]

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced. • Developed a high order finite volume scheme for elliptic interface problems. • Demonstrated high order accuracy and robustness on a variety of tests. • Developed method for computing exact geometric information from implicit function. • Provided general truncation error analysis based on cell-centered Taylor expansions. • Extended advances in Embedded Boundary methods to elliptic interface problem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25,45] , could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes. Highlights • A new class of high order finite difference and finite volume WENO schemes are constructed. • These schemes are based on the multi-resolution idea, and a series of unequal-sized hierarchical central spatial stencils. • These schemes can use arbitrary positive linear weights, and are easy to implement for one and multi-dimensions. • These schemes have a gradual degrading of accuracy near discontinuities. [ABSTRACT FROM AUTHOR]

The icosahedral hexagonal grid is a quasi-uniform discretization of the sphere, generated as the dual of a triangular grid derived by successively refining the faces of an icosahedron embedded in the sphere. This grid contains twelve pentagonal cells and a large number of hexagonal cells, and the bounded area ratio between the smallest and largest cells eliminates polar singularities. This structure, however, makes the derivation of high-order numerical methods more difficult than for grids with a more regular structure. This work progresses towards the goal of high-order operators for the dynamical core of a global atmospheric model by developing a nonstaggered finite volume method on this grid, featuring upwinding applied only when necessary for stability, using an edge-centered reconstruction that directly accommodates the grid's intrinsic curvature. These operators compute the cell-average divergence, curl, and gradient with accuracy between second and fourth order for test functions. When applied to shallow water test cases, the resulting method is between third and fourth order accurate for the transport equation and between second and third order accurate for the full shallow water system. [ABSTRACT FROM AUTHOR]

It is shown that the hyperbolic method, if formulated with diffusive fluxes as additional solution variables, works directly and naturally for diffusion problems with discontinuous coefficients as well as discontinuous tangential fluxes across a material interface. A suitable formulation is given, and numerical examples are shown to demonstrate that a hyperbolic scheme can produce exact non-differentiable piecewise-linear solutions for problems with discontinuous coefficients and tangential fluxes. [ABSTRACT FROM AUTHOR]

In [1] , the authors proposed a charge-conservative numerical framework for simulating electrohydrodynamic two-phase flows where the electric force was discretely treated as the divergence of Maxwell stress tensor because the use of volume-averaged electric force was found to be inaccurate. In this letter, we show that this framework still suffers from inaccuracies, particularly at high permittivity ratios and propose a simple solution that involves reconstruction of electric displacement rather than the electric field. We demonstrate the efficacy of the new remedial approach through simple numerical experiments for different electrical behaviour of the fluids over a range of permittivity ratios. [ABSTRACT FROM AUTHOR]

This work is concerned with spatial advection and diffusion discretization technology within the field of Computational Fluid Dynamics (CFD). In this context, a novel method is proposed, which is dubbed the Enhanced Taylor Advection–Diffusion (ETAD) scheme. The model equation employed for design of the scheme is the scalar advection–diffusion equation, the industrial application being incompressible laminar and turbulent flow. Developed to be implementable into finite volume codes, ETAD places specific emphasis on improving accuracy on stretched structured and unstructured meshes while considering both advection and diffusion aspects in a holistic manner. A vertex-centered structured and unstructured finite volume scheme is used, and only data available on either side of the volume face is employed. This includes the addition of a so-called mesh stretching metric. Additionally, non-linear blending with the existing NVSF scheme was performed in the interest of robustness and stability, particularly on equispaced meshes. The developed scheme is assessed in terms of accuracy – this is done analytically and numerically, via comparison to upwind methods which include the popular QUICK and CUI techniques. Numerical tests involved the 1D scalar advection–diffusion equation, a 2D lid driven cavity and turbulent flow case. Significant improvements in accuracy were achieved, with L 2 error reductions of up to 75%. [ABSTRACT FROM AUTHOR]

We develop a general interface procedure to couple both structured and unstructured parts of a hybrid mesh in a non-collocated, multi-block fashion. The target is to gain optimal computational efficiency in fluid dynamics simulations involving complex geometries. While guaranteeing stability, the proposed procedure is optimized for accuracy and requires minimal algorithmic modifications to already existing schemes. Initial numerical investigations confirm considerable efficiency gains compared to non-hybrid calculations of up to an order of magnitude. [ABSTRACT FROM AUTHOR]

A Finite-Volume scheme for the numerical computations of compressible single- and two-phase flows in flexible pipelines is proposed based on an approximate Godunov-type approach. The spatial discretization is here obtained using the HLLC scheme. In addition, the numerical treatment of abrupt changes in area and network including several pipelines connected at junctions is also considered. The proposed approach is based on the integral form of the governing equations making it possible to tackle general equations of state. A coupled approach for the resolution of fluid-structure interaction of compressible fluid flowing in flexible pipes is considered. The structural problem is solved using Euler–Bernoulli beam finite elements. The present Finite-Volume method is applied to ideal gas and two-phase steam-water based on the Homogeneous Equilibrium Model (HEM) in conjunction with a tabulated equation of state in order to demonstrate its ability to tackle general equations of state. The extensive application of the scheme for both shock tube and other transient flow problems demonstrates its capability to resolve such problems accurately and robustly. Finally, the proposed 1-D fluid-structure interaction model appears to be computationally efficient. [ABSTRACT FROM AUTHOR]

This paper discusses issues encountered by the hyperbolic method for diffusion (Nishikawa, 2007) [1] in dimensional heat conduction problems, and proposes a practical resolution. It is shown that the relaxation length must be scaled by a reference length of a domain of interest for solving dimensional equations, and the corresponding non-dimensionalized length should be given an optimal value for fast iterative convergence. To achieve both, a practical formula is proposed for computing a reference length for a given computational grid, such that ( 2 π ) − 1 gives an optimal value for rectangular domains and also serves as an effective approximation for general domains. Numerical results confirm that the proposed scaling is critically important for rendering hyperbolic diffusion schemes independent of the grid unit and for achieving optimal performance of a hyperbolic diffusion solver. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM). [ABSTRACT FROM AUTHOR]

It is well known that low diffusion Riemann solvers with minimal smearing on contact and shear waves are vulnerable to shock instability problems, including the carbuncle phenomenon. In the present study, we concentrate on exploring where the instability grows out and how the dissipation inherent in Riemann solvers affects the unstable behaviors. With the help of numerical experiments and a linearized analysis method, it has been found that the shock instability is strongly related to the unstable modes of intermediate states inside the shock structure. The consistency of mass flux across the normal shock is needed for a Riemann solver to capture strong shocks stably. The famous carbuncle phenomenon is interpreted as the consequence of the inconsistency of mass flux across the normal shock for a low diffusion Riemann solver. Based on the results of numerical experiments and the linearized analysis, a robust Godunov-type scheme with a simple cure for the shock instability is suggested. With only the dissipation corresponding to shear waves introduced in the vicinity of strong shocks, the instability problem is circumvented. Numerical results of several carefully chosen strong shock wave problems are investigated to demonstrate the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We propose a new sixth-order finite volume scheme to solve the bidimensional linear steady-state Stokes problem on staggered unstructured meshes and complex geometries. The method is based on several classes of polynomial reconstructions to accurately evaluate the diffusive fluxes, the pressure gradient, and the velocity divergence. The main difficulty is to handle the div-grad duality to avoid numerical locking and oscillations. A new preconditioning technique based on the construction of a pseudo-inverse matrix is also proposed to dramatically reduce the computational effort. Several numerical simulations are carried out to highlight the performance of the new method. [ABSTRACT FROM AUTHOR]

In this paper a third order finite volume weighted essentially non-oscillatory scheme is designed for solving hyperbolic conservation laws on tetrahedral meshes. Comparing with other finite volume WENO schemes designed on tetrahedral meshes, the crucial advantages of such new WENO scheme are its simplicity and compactness with the application of only six unequal size spatial stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, and easy choice of the positive linear weights without considering the topology of the meshes. The original innovation of such scheme is to use a quadratic polynomial defined on a big central spatial stencil for obtaining third order numerical approximation at any points inside the target tetrahedral cell in smooth region and switch to at least one of five linear polynomials defined on small biased/central spatial stencils for sustaining sharp shock transitions and keeping essentially non-oscillatory property simultaneously. By performing such new procedures in spatial reconstructions and adopting a third order TVD Runge–Kutta time discretization method for solving the ordinary differential equation (ODE), the new scheme's memory occupancy is decreased and the computing efficiency is increased. So it is suitable for large scale engineering requirements on tetrahedral meshes. Some numerical results are provided to illustrate the good performance of such scheme. [ABSTRACT FROM AUTHOR]

This paper discusses effects of high-frequency damping on iterative convergence of an implicit defect-correction solver for viscous problems. The study targets a finite-volume discretization with a one parameter family of damped viscous schemes. The parameter α controls high-frequency damping: zero damping with α = 0 , and larger damping for larger α ( > 0 ) . Convergence rates are predicted for a model diffusion equation by a Fourier analysis over a practical range of α . It is shown that the convergence rate attains its minimum at α = 1 on regular quadrilateral grids, and deteriorates for larger values of α . A similar behavior is observed for regular triangular grids. In both quadrilateral and triangular grids, the solver is predicted to diverge for α smaller than approximately 0.5. Numerical results are shown for the diffusion equation and the Navier–Stokes equations on regular and irregular grids. The study suggests that α = 1 and 4/3 are suitable values for robust and efficient computations, and α = 4 / 3 is recommended for the diffusion equation, which achieves higher-order accuracy on regular quadrilateral grids. Finally, a Jacobian-Free Newton–Krylov solver with the implicit solver (a low-order Jacobian approximately inverted by a multi-color Gauss–Seidel relaxation scheme) used as a variable preconditioner is recommended for practical computations, which provides robust and efficient convergence for a wide range of α . [ABSTRACT FROM AUTHOR]

We present a new fully-implicit, mixed-hybrid, finite-element (MHFE) discretization scheme for general-purpose compositional reservoir simulation. The locally conservative scheme solves the coupled momentum and mass balance equations simultaneously, and the fluid system is modeled using a cubic equation-of-state. We introduce a new conservative flux approach for the mass balance equations for this fully-implicit approach. We discuss the nonlinear solution procedure for the proposed approach, and we present extensive numerical tests to demonstrate the convergence and accuracy of the MHFE method using tetrahedral elements. We also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability. Finally, we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids. [ABSTRACT FROM AUTHOR]

Including terrain in atmospheric models gives rise to mesh distortions near the lower boundary that can degrade accuracy and challenge the stability of transport schemes. Multidimensional transport schemes avoid splitting errors on distorted, arbitrary meshes, and method-of-lines schemes have a low computational cost because they perform reconstructions at fixed points. This paper presents a multidimensional method-of-lines finite volume transport scheme, “cubicFit”, which is designed to be numerically stable on arbitrary meshes. Stability conditions derived from a von Neumann stability analysis are imposed during model initialisation to obtain stability and improve accuracy in distorted regions of the mesh, and near steeply-sloping lower boundaries. Reconstruction calculations depend upon the mesh only, needing just one vector multiply per face per time-stage irrespective of the velocity field. The cubicFit scheme is evaluated using three, idealised numerical tests. The first is a variant of a standard horizontal transport test on severely distorted terrain-following meshes. The second is a new test case that assesses accuracy near the ground by transporting a tracer at the lower boundary over steep terrain on terrain-following meshes, cut-cell meshes, and new, slanted-cell meshes that do not suffer from severe time-step constraints associated with cut cells. The third, standard test deforms a tracer in a vortical flow on hexagonal–icosahedral meshes and cubed-sphere meshes. In all tests, cubicFit is stable and largely insensitive to mesh distortions, and cubicFit results are more accurate than those obtained using a multidimensional linear upwind transport scheme. The cubicFit scheme is second-order convergent regardless of mesh distortions. [ABSTRACT FROM AUTHOR]

The accurate description and robust simulation at relatively low cost of a size polydisperse population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, moment methods, derived from a population balance equation, represent a very interesting strategy. However, one of the major issues of such methods is the realizability: the numerical schemes have to ensure that the moment sets stay realizable, i.e. that an underlying distribution exists. This issue is all the more crucial that some moment vectors can be at the boundary of the moment space for practical applications, corresponding to a population of particles with only one or a few sizes. It is then investigated here for the advection operator, for which it is particularly significant. Then second order realizable kinetic finite volume schemes are designed, with two strategies for the fluxes evaluation based on the work of Kah et al. [1] and of Vikas et al. [2] , which are here completely revisited, extended to take into account the boundary of the moment space and any number of moments, analyzed and compared in a Cartesian mesh context. For a potential easiest generalization to unstructured meshes, simplified but still realizable versions of these schemes are also developed. The high accuracy of all the schemes is then numerically checked on 1D and 2D test cases, with Cartesian meshes, and their robustness is shown, even when some moment vectors are at the boundary of the moment space. [ABSTRACT FROM AUTHOR]

This paper introduces a two-dimensional cell-centred finite volume discretization of the Poisson problem on adaptive Cartesian quadtree grids which exhibits second order accuracy in both the solution and its gradients, and requires no grading condition between adjacent cells. At T-junction configurations, which occur wherever resolution differs between neighboring cells, use of the standard centred difference gradient stencil requires that ghost values be constructed by interpolation. To properly recover second order accuracy in the resulting numerical gradients, prior work addressing block-structured grids and graded trees has shown that quadratic, rather than linear, interpolation is required; the gradients otherwise exhibit only first order convergence, which limits potential applications such as fluid flow. However, previous schemes fail or lose accuracy in the presence of the more complex T-junction geometries arising in the case of general non-graded quadtrees, which place no restrictions on the resolution of neighboring cells. We therefore propose novel quadratic interpolant constructions for this case that enable second order convergence by relying on stencils oriented diagonally and applied recursively as needed. The method handles complex tree topologies and large resolution jumps between neighboring cells, even along the domain boundary, and both Dirichlet and Neumann boundary conditions are supported. Numerical experiments confirm the overall second order accuracy of the method in the L ∞ norm. [ABSTRACT FROM AUTHOR]

The purpose of this work is to construct a simple, efficient and accurate well-balanced numerical scheme for one-dimensional (1D) blood flow in large arteries with varying geometrical and mechanical properties. As the steady states at rest are not relevant for blood flow, we construct two well-balanced hydrostatic reconstruction techniques designed to preserve low-Shapiro number steady states that may occur in large network simulations. The Shapiro number S h = u / c is the equivalent of the Froude number for shallow water equations and the Mach number for compressible Euler equations. The first is the low-Shapiro hydrostatic reconstruction (HR-LS), which is a simple and efficient method, inspired from the hydrostatic reconstruction technique (HR). The second is the subsonic hydrostatic reconstruction (HR-S), adapted here to blood flow and designed to exactly preserve all subcritical steady states. We systematically compare HR, HR-LS and HR-S in a series of single artery and arterial network numerical tests designed to evaluate their well-balanced and wave-capturing properties. The results indicate that HR is not adapted to compute blood flow in large arteries as it is unable to capture wave reflections and transmissions when large variations of the arteries' geometrical and mechanical properties are considered. On the contrary, HR-S is exactly well-balanced and is the most accurate hydrostatic reconstruction technique. However, HR-LS is able to compute low-Shapiro number steady states as well as wave reflections and transmissions with satisfying accuracy and is simpler and computationally less expensive than HR-S. We therefore recommend using HR-LS for 1D blood flow simulations in large arterial network simulations. [ABSTRACT FROM AUTHOR]

The spatial discretization of convective terms in compressible flow equations is studied from an abstract viewpoint, for finite-difference methods and finite-volume type formulations with cell-centered numerical fluxes. General conditions are sought for the local and global conservation of primary (mass and momentum) and secondary (kinetic energy) invariants on Cartesian meshes. The analysis, based on a matrix approach, shows that sharp criteria for global and local conservation can be obtained and that in many cases these two concepts are equivalent. Explicit numerical fluxes are derived in all finite-difference formulations for which global conservation is guaranteed, even for non-uniform Cartesian meshes. The treatment reveals also an intimate relation between conservative finite-difference formulations and cell-centered finite-volume type approaches. This analogy suggests the design of wider classes of finite-difference discretizations locally preserving primary and secondary invariants. • Criteria for conservation of primary and secondary invariants for compressible flow. • Equivalence exists between global and local conservation; explicit fluxes determined. • Analysis applies to finite-differences and finite-volumes on non-uniform grids. • Relation identified between conservative finite-difference and finite-volume methods. • Energy-preserving methods based on dual-sided discretizations characterized. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel way to represent the interface for two-phase flows with phase change. We combine a level-set method with a Cartesian embedded boundary method and take advantage of both. This is part of an effort to obtain a numerical strategy relying on Cartesian grids allowing the simulation of complex boundaries with possible change of topology while retaining a high-order representation of the gradients on the interface and the capability of properly applying boundary conditions on the interface. This leads to a two-fluid conservative second-order numerical method. The ability of the method to correctly solve Stefan problems, onset dendrite growth with and without anisotropy is demonstrated through a variety of test cases. Finally, we take advantage of the two-fluid representation to model a Rayleigh–Bénard instability with a melting boundary. [ABSTRACT FROM AUTHOR]

We propose a new and simple method to extend any two-fluid dimensionally split VOF schemes on Cartesian meshes to N -fluid problems in 2D and 3D. The method is symmetric by permutation of the fluids, so that it is independent of the ordering of materials, and guarantees natural properties of the volume fractions. The method is called Renormalized VOF or ReVOF , since it relies on a new renormalization algorithm to post-process N independent calls to two-fluid numerical fluxes. Proof that the algorithm finishes is given. Various numerical test cases for advection, rotation and distorsion of three to five fluids in 2D are presented, along with a 3D example. • New method for Volume-Of-Fluid simulation of more than two non-miscible fluids. • Simple post-processing after several independent calls to a two-fluid VOF scheme. • Generic method that can be based on any existing two-fluid VOF schemes. • Proof of respect of the symmetries and the natural properties of volume fractions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. First, using the Cattaneo's relaxation approach, we reformulate the system of governing equations for the pollutant transport by shallow water flows over non-flat topography and anisotropic diffusion as hyperbolic balance laws with stiff source terms. The proposed relaxation system circumvents the infinite wave speed paradox which is inherent in standard advection–diffusion models. This turns out to give a larger stability range for the choice of the time step. Second, following a flux splitting approach, we derive a novel numerical method to discretise the resulting problem. In particular, we propose a new flux splitting and study the associated two systems of differential equations, called the “hydrodynamic” and the “relaxed diffusive” system, respectively. For the presented splitting we analyse the resulting two systems of differential equations and propose two discretisation schemes of the Godunov-type. These schemes are simple to implement, robust, accurate and fast when compared with existing methods. The resulting method is implemented on unstructured meshes and is systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems including non-flat topography and wetting and drying problems. Formal second order accuracy is assessed through convergence rates studies. [ABSTRACT FROM AUTHOR]

In this paper we present a new nonlinear finite volume scheme preserving positivity for diffusion equations. The main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided. Two nonnegative parameters are introduced to define a new nonlinear two-point flux, in which one point is the cell-center and the other is the midpoint of cell-edge. The final flux on the edge is obtained by the continuity of normal flux. Numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes. [ABSTRACT FROM AUTHOR]

Many newly developed climate, weather and ocean global models are based on quasi-uniform spherical polygonal grids, aiming for high resolution and better scalability. Thuburn et al. (2009) and Ringler et al. (2010) developed a C staggered finite volume/difference method for arbitrary polygonal spherical grids suitable for these next generation dynamical cores. This method has many desirable mimetic properties and became popular, being adopted in some recent models, in spite of being known to possess low order of accuracy. In this work, we show that, for the nonlinear shallow water equations on non-uniform grids, the method has potentially 3 main sources of inconsistencies (local truncation errors not converging to zero as the grid is refined): (i) the divergence term of the continuity equation, (ii) the perpendicular velocity and (iii) the kinetic energy terms of the vector invariant form of the momentum equations. Although some of these inconsistencies have not impacted the convergence on some standard shallow water test cases up until now, they may constitute a potential problem for high resolution 3D models. Based on our analysis, we propose modifications for the method that will make it first order accurate in the maximum norm. It preserves many of the mimetic properties, albeit having non-steady geostrophic modes on the f-sphere. Experimental results show that the resulting model is a more accurate alternative to the existing formulations and should provide means of having a consistent, computationally cheap and scalable atmospheric or ocean model on C staggered Voronoi grids. [ABSTRACT FROM AUTHOR]

We propose and analyze finite volume Godunov type methods based on discontinuous flux for a 2 × 2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The scheme is made well-balanced by modifying the flux function locally by including source term as a part of the convection term. Its extension to multi-dimensions is not straightforward for which an approach has been introduced here based on Transport Rays . This approach is compared with another approach for inclusion of source term which uses the idea of inverting the divergence operator relying on the Curl-free component of the Helmholtz decomposition of the source term. Numerical experiments are presented to illustrate the efficiency of the proposed scheme for both unsteady and steady state calculations and to make comparisons with the previously studied finite difference and semi-Lagrangian approaches by Falcone and Finzi Vita. [ABSTRACT FROM AUTHOR]

We present a scalable collocated Finite Volume Method (FVM) to simulate induced seismicity as a result of pore pressure changes. A discrete system is obtained based on a fully-implicit fully-coupled description of flow, elastic deformation, and contact mechanics at fault surfaces on a flexible unstructured mesh. The cell-centered collocated scheme leads to a convenient integration of the different physical equations, as the unknowns share the same discrete locations on the mesh. Additionally, a generic multi-point flux approximation is formulated to treat heterogeneity, anisotropy, and cross-derivative terms for both flow and mechanics equations. The resulting system, though flexible and accurate, can lead to excessive computational costs for field-relevant applications. To resolve this limitation, a scalable processing algorithm is developed and presented. Several proof-of-concept numerical tests, including benchmark studies with analytical solutions, are investigated. It is found that the presented method is indeed accurate and efficient; and provides a promising framework for accurate and efficient simulation of induced seismicity in various geoscientific applications. • We develop a collocated Finite Volume scheme for faulted poroelastic media. • The scheme works on the cell-centered grid, it supports faults, arbitraty material anisotropy and heterogeneity. • Fully coupled multi-point stress and multi-point flux approximations derived from local balance are extended to faulted media. • We validate the scheme in several benchmarks relevant for induced seismicity. • Block-partitioned preconditioning strategy is applied to facilitate the scalability of the developed framework. [ABSTRACT FROM AUTHOR]

• An overset grid ADER scheme is proposed for the Navier-Stokes equations. • The scheme provides an ideal setting to compute flows past moving objects. • A compact transmission condition is devised to limit communication among the grids. • Second-order accuracy is preserved in both space and time. • DOFs are reduces up to two order of magnitude wrt cases from the literature. A space-time Finite Volume method is devised to simulate incompressible viscous flows in an evolving domain. Inspired by the ADER method (based on a Finite-Element-prediction/Finite-Volume-correction approach), the Navier-Stokes equations are discretized onto a space-time overset grid which is able to take into account both the shape of a possibly moving object and the evolution of the domain. A compact transmission condition is employed in order to mutually exchange information from one mesh to the other. The resulting method is second order accurate in space and time for both velocity and pressure. The accuracy and efficiency of the method are tested through reference simulations. [ABSTRACT FROM AUTHOR]

This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraint-preserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten–Lax–van Leer flux, and third-order strong-stability-preserving time discretization. Due to the relativistic effects, the primitive variables (namely, the rest-mass density, velocity, and pressure) are highly nonlinear implicit functions in terms of the conservative variables, making the design and analysis of our method nontrivial. To address the difficulties arising from the strong nonlinearity, we adopt a novel quasilinear technique for the theoretical proof of the PCP property. Three provable convergence-guaranteed iterative algorithms are also introduced for the robust recovery of primitive quantities from admissible conservative variables. We also propose a slight modification to an existing WENO reconstruction to ensure the scaling invariance of the nonlinear weights and thus to accommodate the homogeneity of the evolution operator, leading to the advantages of the modified WENO reconstruction in resolving multi-scale wave structures. Extensive numerical examples are presented to demonstrate the robustness, expected accuracy, and high resolution of the proposed method. • Propose a robust finite volume WENO method for relativistic hydrodynamics on unstructured meshes. • The method is proven to preserve the positivity of pressure/density and the bound of velocity. • Propose three provably convergent iterative algorithms for robust recovery of primitive variables. • Propose a modification to the WENO weights to ensure its scaling invariance for multiscale problems. • Extensive benchmark and challenging tests demonstrate the robustness and effectiveness of the method. [ABSTRACT FROM AUTHOR]

• A 3D finite volume algorithm is developed to handle the dynamics of filaments. • Forces and torques acting between filament cross-sections are automatically balanced. • The method accurately simulates filaments in a large range of different contexts. • Ideal method for handling free filaments. • Force and torque discretization can be directly implemented in other algorithms. Filaments, rods, and beams are ubiquitous in biology and in many man-made products and structures. While a substantial amount of research has been done to understand the statics and dynamics of these long, thin objects, there remain many unanswered and unstudied problems related to the dynamics of bending and twisting filamentary objects. Simulating the general dynamics of these structures in 3D remains challenging. For example, the net force and torque on a free filament immersed in fluid at low Reynolds number must be zero. However, standard finite difference approaches will often fail to preserve the zero force and torque conditions. These numerical artifacts cause spurious rotations and translations that prohibit, or at least limit, their accuracy in simulating the dynamics of filaments, rods, and beams in these contexts (such as the free-swimming motion of a filamentary microorganism). Here we develop a finite volume discretization based on the Kirchoff equations that naturally guarantees the correct total integral of the forces and torques on filaments, rods, or beams. We then couple this discretization to resistive force theory to develop a stable, accurate dynamic algorithm of filament motion at low Reynolds number. We use a range of sample problems to highlight the utility, stability, and accuracy of this method. While our sample problems focus on low Reynolds number dynamics in the context of resistive force theory (RFT), our discretized finite volume algorithm is general and can be applied to inertial dynamics, immersed boundary methods, and boundary integral methods, as well. [ABSTRACT FROM AUTHOR]

In this paper, we construct second- and third-order hyperbolic residual-distribution schemes for general advection–diffusion problems on arbitrary triangular grids. We demonstrate that the accuracy of the second-order hyperbolic schemes in [J. Comput. Phys. 227 (2007) 315–352] and [J. Comput. Phys. 229 (2010) 3989–4016] can be greatly improved by requiring the scheme to preserve exact quadratic solutions. The improved second-order scheme can be easily extended to a third-order scheme by further requiring the exactness for cubic solutions. These schemes are constructed based on the SUPG methodology formulated in the framework of the residual-distribution method, and thus can be considered as economical and powerful alternatives to high-order finite-element methods. For both second- and third-order schemes, we construct a fully implicit solver by the exact residual Jacobian of the proposed second-order scheme, and demonstrate rapid convergence, typically with no more than 10–15 Newton iterations (and about 200–800 linear relaxations per Newton iteration), to reduce the residuals by ten orders of magnitude. We also demonstrate that these schemes can be constructed based on a separate treatment of the advective and diffusive terms, which paves the way for the construction of hyperbolic residual-distribution schemes for the compressible Navier–Stokes equations. Numerical results show that these schemes produce exceptionally accurate and smooth solution gradients on highly skewed and anisotropic triangular grids even for a curved boundary problem, without introducing curved elements. A quadratic reconstruction of the curved boundary normals and a high-order integration technique on curved boundaries are also provided in details. [ABSTRACT FROM AUTHOR]

The objective of this work is to present a conservative coupling method between an inviscid compressible fluid and a deformable structure undergoing large displacements. The coupling method combines a cut-cell Finite Volume method, which is exactly conservative in the fluid, and a symplectic Discrete Element method for the deformable structure. A time semi-implicit approach is used for the computation of momentum and energy transfer between fluid and solid, the transfer being exactly balanced. The coupling method is exactly mass-conservative (up to round-off errors in the geometry of cut-cells) and exhibits numerically a long-time energy-preservation for the coupled system. The coupling method also exhibits consistency properties, such as conservation of uniform movement of both fluid and solid, absence of numerical roughness on a straight boundary, and preservation of a constant fluid state around a wall having tangential deformation velocity. The performance of the method is assessed on test cases involving shocked fluid flows interacting with two and three-dimensional deformable solids undergoing large displacements. [ABSTRACT FROM AUTHOR]

The following work is devoted to the construction and validation of a numerical scheme for the 2D shallow water system on unstructured meshes, supplemented by topography and friction source terms. Approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws, involving the water free surface and some fractions of water, accounting for the topography variations. The discretization of the friction source terms relies on the use of a modified Riemann solver for the flux computation. The resulting scheme is then corrected in order to achieve an asymptotic regime preservation. A MUSCL reconstruction is also performed to increase the space order of accuracy. The overall numerical approach is shown to be consistent, well-balanced and to satisfy a domain invariant principle. These results are assessed through several benchmark tests, involving complex geometry and varying bathymetry. In the presence of dry areas, special interest is given to the wave front speed computation, highlighting the stability of the method, even when implementing the asymptotic preserving correction. [ABSTRACT FROM AUTHOR]

In this study, we propose a charge, momentum, and energy conserving discretization for the 1D–1V Vlasov–Ampère system of equations on an Eulerian grid. The new conservative discretization is nonlinear in nature, but can be efficiently converged with a moment-based nonlinear accelerator algorithm. We demonstrate the conservation and convergence properties of the scheme with various numerical examples, including a multi-scale ion–acoustic shockwave problem. [ABSTRACT FROM AUTHOR]

Numerical schemes for ideal magnetohydrodynamics (MHD) that are based on the standard finite volume method (FVM) exhibit pseudo-convergence in which irregular structures no longer exist only after heavy grid refinement. We describe a method for obtaining solutions for coplanar and near-coplanar cases that consist of only regular structures, independent of grid refinement. The method, referred to as Compound Wave Modification (CWM), involves removing the flux associated with non-regular structures and can be used for simulations in two- and three-dimensions because it does not require explicitly tracking an Alfvén wave. For a near-coplanar case, and for grids with 2 13 points or less, we find root-square-mean-errors (RMSEs) that are as much as 6 times smaller. For the coplanar case, in which non-regular structures will exist at all levels of grid refinement for standard FVMs, the RMSE is as much as 25 times smaller. [ABSTRACT FROM AUTHOR]

We introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green–Naghdi equations over varying topography. These new Green–Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces. For these new models, we develop a high order, well balanced, and robust numerical code relying on a hybrid finite volume and finite difference splitting approach. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through an SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth. Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations. [ABSTRACT FROM AUTHOR]