On the pathological behavior of upwind schemes

Abstract

Despite constant progress in the developement 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 them is the odd-even decoupling that occurs along planar shocks aligned with the mesh. Quirk proposed to test this shortcoming with the propagation of a planar shock in a duct. First, we give a few results on some failings. 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, EFM Pullin's scheme, EIM Macrossan's scheme and AUSM Liou's scheme. Strict stability is shown to be desirable to avoid must of these flaws.