Report 1996-04

Boundary Treatment for a Cartesian Grid Method

H. Forrer

Abstract: We are interested in a numerical solution to the Euler Equations in complicated 2-dimensional geometries using a Cartesian grid method. To avoid stability problems or loss of accuracy along the boundary, this requires a special treatment of the irregular cells along the boundary. In this paper we present a new technique for the boundary treatment. The technique is built upon a high resolution finite volume method with dimensional splitting. To avoid stability problems for small boundary cells due to instable fluxes, we use an enlargement of the domain of dependence. The enlarged domains may lie beyond the boundary. By a local mirroring at the boundary we determine values of the flow variables also for these regions. This enables us to calculate stable fluxes for the small boundary cells. These fluxes are formally of second order accuracy.

Among other examples we calculate a Prandtl-Meyer expansion, a double Mach reflection and a shock diffraction by a pair of cylinders. The latter example points to a major advantage of Cartesian grid methods, the ability to cope with complicated geometries.

Paper: Available as PDF (1 MB) or as hardcopy to order reports@sam.math.ethz.ch.