Extension to non-uniform meshes of a high order computationally explicit kinetic scheme for hyperbolic conservation laws

Abstract

In this paper, we present an extension to non-uniform meshes of a 1D scheme [Rémi Abgrall and Davide Torlo. “Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme”. In: Abgrall and Torlo (2022). This scheme is arbitrary high order convergent in space and time for any hyperbolic system of conservation laws. It is based on a Finite Difference technique. We show that this numerical method is not conservative but it satisfies a Lax–Wendroff theorem under restrictive conditions on the mesh. To relax this condition we propose a Finite Volume alternative. This new discretization can be seen as a direct generalization to non-uniform meshes of the Finite Difference schemes in the sense that the fluxes of both methods are the same on uniform meshes. We apply the two schemes to the Euler system and we assess their performances on regarding test problems of the literature.