We present a novel relaxation scheme for the simulation of compressible material flows at all speeds. An Eulerian model describing gases, fluids and elastic solids with the same system of conservation laws is adopted. The proposed numerical scheme is based on a fully implicit time discretization, easily implemented thanks to the linearity of the transport operator in the relaxation system. The spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in all different Mach regimes. The scheme is validated by simulating gas and liquid flows with different Mach numbers. The simulation of the deformation of compressible solids is also approached, assessing the ability of the scheme in approximating material waves in hyperelastic media.

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