A HIGH-RESOLUTION EULER SOLVER BASED ON MULTIGRID, SEMI-COARSENING, AND DEFECT CORRECTION

In an earlier paper, an O(N) method for the computation of stationary solutions to the Euler equations of inviscid compressible gas dynamics has been described. The method is a variant of the multigrid technique and employs semi-coarsening in all co-ordinate directions simultaneously. It provides good convergence rates for first-order upwind discretisations even in the case of alignment, the flow being aligned with the grid. Here we discuss the application of this scheme to higher-order discretisations. Two-grid analysis for the linear constant-coefficient case shows that it is difficult to obtain uniformly good convergence rates for a higher-order scheme, because of waves perpendicular to stream lines. The defect correction technique suffers from the same problem. However, convergence to a point where the residual of the total error (the sum of the iteration error and the discretisation error) is of the order of the truncation error can be obtained in about seven defect correction cycles, according to estimates for the linear constant-coefficient equations. This result is explored for the nonlinear case by some illustrative numerical experiments. © 1992.