  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.
