Efficient multigrid computation of steady hypersonic flows

In steady hypersonic flow computations, Newton iteration as a local relaxation procedure and nonlinear multigrid iteration as an acceleration procedure may both easily fail. In the present chapter, same remedies are presented for overcoming these problems. The equations considered are the steady, two-dimensional Navier-Stokes equations. The equations are discretized by an upwind finite volume method. Collective point Gauss-Seidel relaxation is applied as the standard smoothing technique. In hypersonics this technique easily diverges. First, collective line Gauss-Seidel relaxation is applied as an alternative smoothing technique. Though promising, it also fails in hypersonics. Next, collective point Gauss-Seidel relaxation is reconsidered and improved; a divergence monitor is introduced and in case of divergence a switch is made to a local explicit time stepping technique. Satisfactory singlegrid convergence results are shown for the computation of a hypersonic reentry flow around a blunt forebody with canopy. Unfortunately, with this improved smoothing technique, standard nonlinear multigrid iteration still fails in hypersonics. The robustness improvements made therefore to the standard nonlinear multigrid method are a local damping of the restricted defect, a global upwind prolongation of the correction and a gIobal upwind restriction of the defect. Satisfactory multigrid convergence results are shown for the computation of a hypersonic launch and reentry flow around a blunt forebody with canopy. For the test cases considered, it appears that the improved multigrid method performs significantly better than a standard nonlinear multigrid method. For the test cases considered it appears that the most significant improvement comes from the upwind prolongation, rather than from the upwind restriction and the defect damping