A method for avoiding the acoustic time step restriction in compressible flow

We propose a novel method for alleviating the stringent CFL condition imposed by the sound speed in simulating inviscid compressible flow with shocks, contacts and rarefactions. Our method is based on the pressure evolution equation, so it works for arbitrary equations of state, chemical species etc, and is derived in a straightforward manner. Similar methods have been proposed in the literature, but the equations they are based on and the details of the methods differ significantly. Notably our method leads to a standard Poisson equation similar to what one would solve for incompressible flow, but has an identity term more similar to a diffusion equation. In the limit as the sound speed goes to infinity, one obtains the Poisson equation for incompressible flow. This makes the method suitable for two-way coupling between compressible and incompressible flows and fully implicit solid-fluid coupling, although both of these applications are left to future work. We present a number of examples to illustrate the quality and behavior of the method in both one and two spatial dimensions, and show that for a low Mach number test case we can use a CFL number of 300 (whereas previous work was only able to use a CFL number of 3 on the same example).