Conclusions Some modiŽ cations for the restriction and prolongation process of multigrid methods have been presented that are necessary for the simulation of complex supersonic  owŽ elds. Damping of the restricted residual error and blending between a second-order central and a Ž rst-order upwind prolongation enable convergence in supersonic  ows and resulted in strong reductions in CPU time. The advantageof this kind of blending is the reduction to Ž rst-order prolongationat shock waves only.