12–16 déc. 2022
IMT
Fuseau horaire Europe/Paris

Entropy stable, positivity preserving and well-balanced Godunov-type schemes for multidimensional hyperbolic system of equations

16 déc. 2022, 10:50
45m
Amphithéâtre Schwartz (IMT)

Amphithéâtre Schwartz

IMT

Université Paul Sabatier, 118 Route de Narbonne, 31000 Toulouse France

Orateur

Raphaël LOUBERE

Description

A class of cell centered Finite Volume schemes has been introduced to discretize the equations of
Lagrangian hydrodynamics on moving mesh [4]. In this framework, the numerical fluxes are evaluated
by means of an approximate Riemann solver, based at the nodes of the mesh, which provides the
nodal velocity required to move the mesh in a compatible manner. In this presentation, we describe
the generalization of this type of discretization to hyperbolic systems of conservation laws written in
Eulerian representation. The evaluation of the numerical fluxes relies on an approximate Riemann
solver located at the mesh nodes. The construction of this nodal solver uses the Lagrange-to-Euler
transformation introduced by Gallice [3] and revisited in [1,2] to build positive and entropic Eulerian
Riemann solvers from their Lagrangian counterparts. The application of this formalism to the case of
gas dynamics provides a positive and entropic finite volume scheme under an explicit condition on the
time step. In this work we extend this scheme to handle source terms, in particular for the Shallow
Water system of equations. We show how to render the scheme well-balanced in 1D and 2D. The
numerical assessment of this scheme by means of representative test cases will be presented for the first
and second orders. In particular the good behavior is illustrated by the absence of the typical numerical
pathology of traditional Finite Volume approaches for such system of PDEs.

References
[1] A. Chan, G. Gallice, R. Loubère and P.-H Maire, Positivity preserving and entropy consistent ap-
proximate Riemann solvers dedicated to the high-order MOOD-based Finite Volume discretiza-
tion of Lagrangian and Eulerian gas dynamics. Computers & Fluids, 2021, 229.
[2] G. Gallice, A. Chan, R. Loubère and P.-H Maire, Entropy stable and positivity preserving
Godunov-type schemes for multidimensional hyperbolic systems on unstructured grid. J. Com-
put. Phys, 2022, 468.
[3] G. Gallice, Positive and Entropy Stable Godunov-Type Schemes for Gas Dynamics and MHD
Equations in Lagrangian or Eulerian Coordinates. Numer. Math., 2003, 94.
[4] P.-H Maire, A high-order cell-centered Lagrangian scheme for two-dimensional compressible
fluid flows on unstructured meshes. J.Comput. Phys., 2009, 228.

Documents de présentation

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