Robustness versus accuracy in shock-wave computations

Despite constant progress in the development of upwind schemes, some failin gs still remain. Quirk recently reported (Quirk JJ. A contribution to the g reat Riemann solver debate. International Journal for Numerical Methods in Fluids 1994; 18: 555-574) that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such faili ngs. One of these is the odd-even decoupling that occurs along planar shock s aligned with the mesh. First, a few results on some failings are given, n amely the carbuncle phenomenon and the kinked Mach stem. Then, following Qu irk's analysis of Roe's scheme, general criteria are derived to predict the odd-even decoupling. This analysis is applied to Roe's scheme (Roe PL, App roximate Riemann solvers, parameters vectors, and difference schemes, Journ al of Computational Physics 1981; 43: 357-372), the Equilibrium Flux Method (Puliin DI, Direct simulation methods for compressible inviscid ideal gas flow, Journal of Computational Physics 1980; 34: 231-244), the Equilibrium Interface Method (Macrossan MN, Oliver. RT, A kinetic theory solution metho d for the Navier-Stokes equations, International Journal for Numerical Meth ods in Fluids 1993; 17: 177-193) and the AUSM scheme (Liou MS, Steffen CJ, A new flux splitting scheme, Journal of Computational Physics 1993; 107: 23 -39). Strict stability is shown to be desirable to avoid most of these flaw s. Finally, the link between marginal stability and accuracy on shear waves is established. Copyright (C) 2000 John Wiley & Sons, Ltd.