An asymptotic-preserving all-speed scheme for fluid dynamics and nonlinear elasticity

An implicit relaxation scheme is derived for the simulation of multidimensional flows at all Mach numbers, ranging from very small to order unity. An analytical proof of the asymptotic-preserving property is proposed and the divergence-free condition on the velocity in the incompressible regime is respected. The scheme possesses a general structure, which is independent of the considered state law and thus can be adopted to solve gas and fluid flows, but also deformations of elastic solids. This is achieved by adopting the Jin-Xin relaxation technique in order to get a linear transport operator. The spatial derivatives are thus independent of the equation of state and an easy implementation of fully implicit time discretizations is possible. Several validations on multidimensional tests are presented, showing that the correct numerical viscosity is recovered in both the fully compressible and the low Mach regimes. An algorithm to perform grid adaptivity is also proposed, via the computation of the entropy residual of the scheme.