Semi-implicit relaxed finite volume schemes for hyperbolic multi-scale systems of conservation laws

Abstract

In this paper a new semi-implicit relaxation scheme for the simulation of multi-scale hyperbolic conservation laws based on a Jin-Xin relaxation approach is presented. It is based on the splitting of the flux function into two or more subsystems separating the different scales of the considered model whose stiff components are relaxed thus yielding a linear structure of the resulting relaxation model on the relaxation variables. This allows the construction of a linearly implicit numerical scheme, where convective processes are discretized explicitly. Thanks to this linearity, the discrete scheme can be reformulated in linear decoupled wave-type equations resulting in the same number of evolved variables as in the original system. To obtain a scale independent numerical diffusion, centred fluxes are applied on the implicitly treated terms, whereas classical upwind schemes are applied on the explicit parts. The numerical scheme is validated by applying it on the Toro & V\'azquez-Cend\'on (2012) splitting of the Euler equations and the Fambri (2021) splitting of the ideal MHD equations where the flux is split in two, respectively three sub-systems. The performance of the numerical scheme is assessed running benchmark test-cases from the literature in one and two spatial dimensions.