Robustness versus accuracy in shock-wave computations

Abstract

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd – even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk’s analysis of Roe’s scheme, general criteria are derived to predict the odd – even decoupling. This analysis is applied to Roe’s scheme, the Equilibrium Flux Method, the Equilibrium Interface Method and the AUSM scheme. Strict stability is shown to be desirable to avoid most of these flaws. Finally, the link between marginal stability and accuracy on shear waves is established.