Multigrid solution of the steady Euler equations

A multigrid (MG) method for the approximation of steady solutions to the full two-dimensional Euler equations is described. The space discretization is obtained by the finite volume technique and Osher's approximate Riemann-solver. Symmetric Gauss-Seidel relaxation is applied to solve the nonlinear discrete system of equations. A multigrid method, the full approximation scheme, accelerates this iterative process. In two-dimensional test problems, (subsonic, transonic and supersonic) the multigrid iteration is applied to an initial estimate obtained by the FMG-technique (nested iteration). For the discretization on the different levels, a fully consistent sequence of nested discretizations is used. The prolongations and restrictions selected agree with this consistency. It turns out that the total amount of work required to obtain a solution that is accurate up to truncation error, corresponds to a small number of nonlinear Gauss-Seidel iterations. In transonic flow the rate of convergence of the MG-iteration appears independent of the number of cells in the discretization.