Multidimensional dissipation for upwind schemes: Stability and applications to gas dynamics

In this work, we identify an instability in strictly upwind finite difference schemes when they are applied to the Euler equations in more than one space dimension. We suggest that the well known carbuncle phenomenon is a manifestation of this instability. The usual dimension by dimension extension of one-dimensional upwind schemes to the multidimensional equations of gas dynamics often yields poorly resolved stationary (or slowly moving) shocks when applied to high Mach number grid aligned flows on structured grids. Through linear analysis, we show that this failure is an instability which is the result of inadequate crossflow dissipation offered by strictly upwind schemes. In addition, we offer a new parameter free and easy to implement multidimensional, upwind dissipation modification that provides sufficient crossflow dissipation to eliminate the instability. This new approach is applied to the problem of simulating a three-dimensional, axisymmetric, hypersonic, chemically reacting air flow typically encountered during spacecraft reentry. (C) 1998 Academic Press.