Implicit relaxed all Mach number schemes for gases and compressible materials

We present an implicit relaxation scheme for the simulation of compressible flows in all Mach number regimes based on a Jin Xin relaxation approach. The main features of the proposed scheme lie in its simplicity and effectiveness. Thanks to the linearity of the flux in the relaxation system, the time-semi discrete scheme can be reformulated in linear decoupled elliptic equations resulting in the same number of unknowns as in the original system. To obtain the correct numerical diffusion in all Mach number regimes, a convex combination of upwind and centred fluxes is applied. The numerical scheme is validated by applying it on a Eulerian model for non-linear elasticity. Simulations of gas and fluid flows, as well as deformations of compressible solids are carried out to assess the performance of the numerical scheme in accurately approximating material waves in different Mach regimes.