Linearly implicit all Mach number shock capturing schemes for the Euler equations

We propose a family of simple second order accurate schemes for the numerical solution of Euler equation of gas dynamics that are (linearly)implicit in the acoustic waves, eliminating the acoustic CFL restriction on the time step. The general idea is that explicit differential operators in space relative to convective or material speeds are discretised by upwind schemes or local Lax-Friedrics fluxes and the linear implicit operators, pertaining to acoustic waves, are discretised by central differences. We have compared the results of such schemes on a series of one and two dimensional test problems including classical shock tube configurations. Also we have considered low-Mach number acoustic wave propagation tests as well as nozzle flows in various Mach regimes. The results show that these schemes do not introduce excessive numerical dissipation at low Mach number providing an accurate solution in such regimes. They perform reasonably well also when the Mach number are not too small.