Low-Diffusion Rotated Upwind Schemes, Multigrid and Defect Corrections forSteady, Multi-Dimensional Euler Flows

摘要

Two simple, multidimensional upwind discretizations for the steady Eulerequations are derived, with the emphasis on both accuracy and solvability. The multidimensional 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 does not have 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 within 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 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 standard, grid aligned upwind schemes. The grid decoupled results obtained are promising.