A robust multigrid algorithm for the Euler equations with local preconditioning and semi-coarsening

A semi-coarsened multigrid algorithm with a point block Jacobi, multi-stage smoother for second-order upwind discretizations of the two-dimensional Euler equations which produces convergence rates independent of grid size for moderate subsonic Mach numbers is presented. By modification of this base algorithm to include local preconditioning for low Mach number flows, the convergence becomes largely independent of grid size and Mach number over a range of flow conditions from nearly incompressible to transonic flows, including internal and external flows. A local limiting technique is introduced to increase the robustness of preconditioning in the presence of stagnation points. Computational timings are made showing that the semi-coarsening algorithm requires O(N) time to lower the fine grid residual six orders of magnitude, where N is the number of cells. By comparison, the same algorithm applied to a full-coarsening approach requires O (N-3/2) time, and, in nearly all cases, the semi-coarsening algorithm is faster than full coarsening with the computational savings being greatest on the finest grids. (C) 1999 Academic Press.