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In this paper, we propose an adaptive mesh refinement method for 2D multi-material compressible nonviscous flows in semi-Lagrangian coordinates. The mesh adaptation procedure is local and relies on discrete metric field evaluation. The remapping method is second-order accurate and we prove its stability. We propose a multi-material treatment using two ingredients: the local remeshing is performed in a way to reduce as much as possible mixture creation and an interface reconstruction method is used to avoid material diffusion in mixing cells. The obtained method is almost Lagrangian and can be implemented in a parallel framework. We provide some numerical tests which attest the validity of the method and its robustness.

International audience; Two formulations of the Lax-Wendroff scheme are proposed in this paper for its extension to elastic-plastic solids. The unknown vector consists of the strain or stress components in addition to the velocity ones according to the chosen formulation. The Lax-Wendroff scheme is here implemented in its Richtmyer two-step version so that the projection of the elastic trial stresses onto the yield locus is performed twice per time step. It is shown that it allows to reduce the number of cells required for the approximation of discontinuous plastic waves with respect to a classical one-step strategy consisting in an a posteriori projection of the elastic trial stresses onto the yield locus. Elastic-plastic associated and non-associated constitutive models under small strains are considered in examples. In particular, dynamic ratchetting is simulated with the Armstrong-Frederick nonlinear kinematic hardening. Comparisons with respect to finite element numerical solutions show good agreements.

This article is devoted to the design of importance sampling method for the Monte Carlo simulation of a linear transport equation. This model is of great importance in the simulation of inertial confinement fusion experiments. Our method is restricted to a spherically symmetric idealized design : an outer sphere emitting radiation towards an inner sphere, which in practice should be thought of as the hohlraum and the fusion capsule, respectively. We compute the importance function as the solution of the corresponding stationary adjoint problem. Doing so, we have an important reduction of the variance (by a factor 50 to 100), with a moderate increase of computational cost (by a factor 2 to 8)., Journal of Computational Physics, Elsevier, A Para{\^i}tre

International audience; We introduce a robust and high order strategy to perform the reinitialization in a level set framework. The reinitialization by closest-points (RCP) method is based on geometric considerations. It relies on a gradient descent to find the closest points at the interface in order to solve the eikonal equation and thus reinitializing the level set field. Furthermore, a new algorithm, also based on a similar geometric approach, is introduced to detect precisely all the ill-defined points of the level set. These points, also referred to as kinks, can mislead the gradient descent and more widely impact the accuracy of level set methods. This algorithm coupled with the precise computation of the closest points of the interface, permits the novel method to be robust and accurate even when performing the reinitialization every time step after solving the advection equation. Furthermore, they both require very few given parameters with the advantage of being based on a geometrical approach and independent of the application. The proposed method was tested on various benchmarks, and demonstrated equivalent or even better results compared to solving the Hamilton-Jacobi equation.

This paper deals with the development of efficient incomplete multidimensional Riemann solvers for hyperbolic systems. Departing from a four-waves model for the speeds of propagation arising at each vertex of the computational structured mesh, we present a general strategy for constructing genuinely multidimensional Riemann solvers, that can be applied for solving systems including source and coupling terms. In particular, a simple version of a well-balanced 2d HLL scheme is presented, which is later taken as a basis to build a general class of incomplete Riemann solvers, the so-called Approximate Viscosity Matrix (AVM) schemes. The great advantage of the AVM strategy is the possibility to control the amount of numerical diffusion considered for each hyperbolic system at an affordable computational cost. The presented numerical schemes are shown to be linearly L ∞ -stable for a CFL number up to unity. Our schemes can be used as building blocks for constructing high-order schemes. In this work, a second-order scheme is constructed by using a predictor-corrector MUSCL-Hancock procedure. To test the performances of the proposed schemes, a number of challenging numerical experiments in one-layer and two-layer shallow water systems have been run. The presence of the bottom topography and the coupling terms represent an additional difficulty, that has been solved by reformulating the problem within the path-conservative framework. Finally, the 2d schemes have been shown to be more efficient than their projected 1d×1d counterparts.

In this paper, we propose a two-dimensional nonlinear "multiring" model for blood flow in axisymmetric elastic arteries. It is designed to overcome the numerical difficulties of three-dimensional fluid-structure interaction simulations of blood flow without using the oversimplifications necessary to obtain one-dimensional models of blood flow. This multiring model is derived by integrating over concentric rings of fluid the simplified long-wave Navier-Stokes equations coupled to an elastic model of the arterial wall. The resulting system of balance laws provides a unified framework in which both the motion of the fluid and the displacement of the wall are dealt with simultaneously. The mathematical structure of the multiring model allows us to use a finite volume method that guarantees the conservation of mass and the positivity of the numerical solution and can deal with nonlinear flows and large deformations of the arterial wall. We show that the finite volume numerical solution of the multiring model provides at a reasonable computational cost an asymptotically valid description of blood flow velocity profiles and other averaged quantities (wall shear stress, flow rate, ...) in large elastic and quasi-rigid arteries. In particular, we validate the multiring model against well-known solutions such as the Womersley or the Poiseuille solutions as well as against steady boundary layer solutions in quasi-rigid constricted and expanded tubes.

International audience; A new model for interface problems with capillary effects in compressible fluids is presented together with a specific numerical method to treat capillary flows and pressure waves propagation. This new multiphase model is in agreement with physical principles of conservation and respects the second law of thermodynamics. A new numerical method is also proposed where the global system of equations is split into several submodels. Each submodel is hyperbolic or weakly hyperbolic and can be solved with an adequate numerical method. This method is tested and validated thanks to comparisons with analytical solutions (Laplace law) and with experimental results on droplet breakup induced by a shock wave.

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. Two new hydrostatic reconstruction techniques are introduced for blood flow.Both preserve low-Shapiro steady states relevant for blood flow.Both accurately capture wave reflections for arbitrary large impedance variations.Capturing wave reflections is necessary for large pathological network simulations.

International audience; Moment-of-Fluid (MoF) is a piecewise linear interface reconstruction method that tracks fluid through its volume fraction and centroid, which are deduced from the zeroth and first moments. We present a method that replaces the original minimization stage by an analytic reconstruction algorithm on bi-dimensional Cartesian grids. This algorithm provides accurate results for a lower computational cost than the original minimization algorithm. When more than two fluids are involved, this algorithm can be used coupled with the minimization algorithm. Although this paper deals with Cartesian grids, everything remains valid for any meshes that are made of rectangular cells.

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.

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units., Comment: Submitted to CMAME

International audience; Recently, several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation, avoiding the resolution of a Pois-son equation. These methods can be seen as various levels of approximation of the compressible Navier-Stokes equation in the low Mach number limit. We study the simulation of incompressible wall-bounded flows using several artificial pressure equations in order to determine the most relevant approximations. The simulations are stable using a finite difference method in a staggered grid system, even without dif-fusive term, and converge to the incompressible solution, both in direct numerical simulations and for coarser meshes, to be used in large-eddy simulations. A pressure equation with a convective and a diffusive term produces a more accurate solution than a compressible solver or methods involving more approximations. This suggests that it is near to an optimal level of approximation. The presence of a convective term in the pressure evolution equation is in particular crucial for the accuracy of the method. The rate of convergence of the solution in terms of artificial Mach number is studied numerically and validates the theoretical quadratic convergence rate. We demonstrate that this property can be used to accelerate the rate of convergence using an extrapolation in terms of artificial Mach number. Since the approach is based on an explicit and local system of equations, the numerical procedure is massively parallelisable and has low memory requirements.

International audience; We describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids. The schemes are obtained by properly adapting the explicit construction proposed in Ricchiuto and Abgrall (2010) . In particular, previous work on well balanced integration (Ricchiuto, 2011 ) and preservation of the depth non-negativity (Ricchiuto and Bollermann, 2009 ) is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation. The paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria, and how to obtain a super-consistent approximation for smooth non-trivial moving equilibria. The treatment of the wetting/drying interface is also discussed, giving formal conditions for the preservation of the non-negativity of the depth for a particular case, based on a nonlinear variant of a Lax–Friedrichs type scheme. The approach is analyzed and tested thoroughly. The quality of the numerical results shows the interest in the proposed approach over previously proposed schemes, in terms of accuracy and efficiency.

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.

The interest on embedded boundary methods increases in Computational Fluid Dynamics (CFD) because they simplify then mesh generation problem in the case of the Navier-Stokes equations. The same simplifications occur for the simulation of multi-physics flows, the coupling of fluid-solid interactions in situation of large motions or deformations, to give a few examples. Nevertheless an accurate treatment of the wall boundary conditions remains an issue of the method. In this work, the wall boundary conditions are easily taken into account through a penalization technique, and the accuracy of the method is recovered using mesh adaptation, thanks to the potential of unstructured meshes. Several classical examples are used used to demonstrate that claim.; L'intèrêt, en CFD, pour les méthodes employant des frontières immergées, va croissant car elles simplifient les procèdures de génération de maillage, en particulier dans le cas des équations de Navier Stokes. On peut faire le même constat dans le cas des problèmes multi-physique, comme, par exemple, pour le couplage fluide-structure, quand il y a de très grandes déformations du maillage. Néanmoins, un traitement précis des conditions de parois reste un problème difficile. Dans ce travail, les conditions d eparoi sont prises en compte par une méthode de pénalisation, et la précision est obtenue grâce à une méthode d'adaptation de maillage. En cela, on utilise le potentiel des mailalges non strcturés. Plusieurs exemples classiques de simulations sont proposées pour valider l'approche.

International audience; A residual-based (RB) scheme relies on the vanishing of residual at the steady-state to design a transient first-order dissipation, which becomes high-order at steady-state. Initially designed within a finite-difference framework for computations of compressible flows on structured grids, the RB schemes displayed good convergence, accuracy and shock-capturing properties which motivated their extension to unstructured grids using a finite volume (FV) method. A second-order formulation of the FV–RB scheme for compressible flows on general unstructured grids was presented in a previous paper. The present paper describes the derivation of a third-order FV–RB scheme and its application to hyperbolic model problems as well as subsonic, transonic and supersonic internal and external inviscid flows.

International audience; This article presents numerical analysis and practical considerations for three-dimensional flow computation using an implicit immersed boundary method. The Euler equations, or half a step of the Navier-Stokes equations when using fractional step algorithms, are investigated in their vorticity formulation. The context of flow computation around an arbitrarily shaped body is especially investigated. In conventional immersed boundary methods using vorticity, singular vortex are dispatched over the body surface. In the present study, one prefers using sources of potential velocity field, dispatched on the body, whose nature is not vorticity. Such a formulation is compatible to the Euler equations. In practice, these sources of potential flow produce a velocity through this surface, aiming in practice at cancelling a flow-through velocity. This article focuses on the use of the source-to-flow-through linear application, its properties being the key points for fast convergence. Its self-adjointness, or lack thereof, conditioning and preconditioning aspects are investigated. It follows that computing a velocity field with no-flow-through conditions in complex geometry, when using the source-to-flow-through linear application, can be achieved for 4/3 of the computational cost of standard Poisson equation in a Cartesian box. The robustness of immersed boundaries is especially interesting when used together with vortex-in-cell methods, well known for their robustness in time and their ability to compute accurately convective effects. A few examples, based on real-world geometries, illustrate the method capabilities.

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.

The latest version of the XTOR code which solves a set of the extended magnetohydrodynamic (MHD) equations in toroidal geometry is presented. The numerical method is discussed with particular emphasis on critical issues leading to numerical stability and robustness. This includes the time advance algorithm, the choice of variables and the boundary conditions. The physics in the model includes resistive MHD, anisotropic thermal diffusion and some neoclassical effects. The time advance method used in XTOR is unconditionally stable for linear MHD. First, both the ideal and the resistive MHD parts of the equations are advanced semi-implicitly and then the thermal transport part full-implicitly, using sub-stepping [H. Lutjens, Comp. Phys. Commun. 164 (2004) 301]. The time steps are only weakly limited by the departure of the nonlinear MHD dynamics from the linear one and are automatically defined by a set of nonlinear stability criteria. The robustness of the method is illustrated by some numerically difficult simulations, i.e. sawtooth simulations, the nonlinear destabilization of ballooning instabilities by an internal kink, and the dynamics of a neoclassical tearing mode in International Thermonuclear Experimental Reactor (ITER) [R. Aymar, V.A. Chuyanov, M. Huguet, et al., Nucl. Fusion 41 (2001) 1301] like geometry about its nonlinear stability threshold.

A method is proposed to estimate the velocity field of an unsteady flow using a limited number of flow measurements. The method is based on a non-linear low-dimensional model of the flow and on an expansion of the velocity field in terms of empirical basis functions. The main idea is to impose that the coefficients of the modal expansion of the velocity field gives the best approximation of the available measurements, while at the same time satisfying the non-linear low-order model as closely as possible. Practical applications may range from feedback flow control to the monitoring of the flow in non-accessible regions. The proposed technique is applied to the flow around a confined square cylinder, both in two- and three-dimensional flow regimes. Comparisons are provided with existing linear and non-linear estimation techniques.

International audience; In this paper, we consider a class of explicit marching schemes first proposed in [1] for solving the wave equation in complex geometry using an embedded Cartesian grid. These schemes rely on an integral evolution formula for which the numerical domain of dependence adjusts automatically to contain the true domain of dependence. Here, we refine and analyze a subclass of such schemes, which satisfy a condition we refer to as strong u-consistency. This requires that the evolution scheme be exact for a single-valued approximation to the solution at the previous time steps. We provide evidence that many of these strongly u-consistent schemes are stable and converge at very high order even in the presence of small cells in the grid, while taking time steps dictated by the uniform grid spacing.

International audience; We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48, 235 (1995)] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.

In this paper we examine the application of a domain decomposition method to the solution of Poisson and Helmholtz equations often involved in the oceanic models. By studying a theoretical convergence rate of the method in the case of two subdomains (which can be generalized to some extent to the N-subdomains case), we can link some numerical properties of the method to physical features of the model ocean circulation. In particular, the peculiar role of the barotropic mode with respect to baroclinic modes is discussed. The tuning of the different parameters involved in a practical implementation of this Schwarz alternating method is also addressed for the case of a quasi-geostrophic oceanic model. Some CPU-timings are presented. On 16 processors of a CRAY T3D, the parallel code runs as fast as the sequential version does on a CRAY C90.

International audience; A domain decomposition method which couples a high and a low-fidelity model is proposed to reduce the computational cost of a flow simulation. This approach requires to solve the high-fidelity model in a small portion of the computational domain while the external field is described by a Galerkin-free Proper Orthogonal Decomposition (POD) model. We propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models. This zonal approach can be used to study multi-body configurations or to perform detailed local analyses in the framework of shape optimisation problems. The efficiency of the method to perform predictive low-cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem.