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### 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.