Low-Diffusion Rotated Upwind Schemes, Multigrid and Defect Correction for Steady, Multi-Dimensional Euler Flows

Abstract

Two simple, multi-dimensional upwind discretizations for the steady Euler equations are derived, with the emphasis lying on both a good accuracy and a good solvability. The multi-dimensional upwinding consists of applying a one-dimensional Riemann solver with a locally rotated left and right state, the rotation angle depending on the local flow solution. First, a scheme is derived for which smoothing analysis of point Gauss-Seidel relaxation shows that despite its rather low numerical diffusion, it still enables a good acceleration by multigrid. Next, a scheme is derived which has not any numerical diffusion in crosswind direction, and of which convergence analysis shows that its corresponding discretized equations can be solved efficiently by means of defect correction iteration with in the inner multigrid iteration the first scheme. For the steady, two-dimensional Euler equations, numerical experiments are performed for some supersonic test cases with an oblique contact discontinuity. The numerical results are in good agreement with the theoretical predictions. Comparisons are made with results obtained by a standard, grid-aligned upwind scheme. The grid-decoupled results obtained are promising.