A nonlinear multigrid method for the efficient solution of the steady Euler equations

Abstract

An efficient iterative method has been developed for the accurate solution of the non-isenthalpic steady Euler equations for inviscid flow.

First, the system of conservation laws is space-discretized by a first order finite-volume Osher-discretization. Without time stepping, the steady equations are solved by iteration with nonlinear multiple grid cycles, where a Symmetric Gauss-Seidel method is used as a relaxation . Initial estimates are obtained by the Full Multigrid method. In the pointwise relaxation, the equations corresponding to each cell are kept in block-coupled form, i.e. a Collective Symmetric Gauss-Seidel relaxation is used. In this relaxation local linearization of the equations and the boundary conditions is applied, and one (or a few) step(s) of a Newton iteration is (are) used for the approximate solution of these small nonlinear systems. The first order Osher-discretization has many good properties which foster the efficiency of multigrid iteration. It appears that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (1 to 3) iteration cycles (3 to 8 work units).

To obtain higher accuracy, we use second order finite volume schemes (e.g. the newly developed superbox scheme [ 3 ]), again based on Osher's approximate Riemann solver. The more accurate discretizations are less stable, and hence harder to solve by relaxation iteration. Therefore, we make use of the fact that the solution of the first order scheme can be computed very efficiently, and we solve the second order system (up to truncation error) by one or a few cycles of a defect correction process.