A multigrid method for steady Euler equations based on polynomial flux-difference splitting

Abstract

A flux-difference splitting for steady Euler equations based on the polynomial character of the flux vectors is introduced. The splitting is applied to vertex-centered finite volumes. A discrete set of equations is obtained which is both conservative and positive. The splitting is done in an algebraically exact way, so that shocks are represented as sharp discontinuities, without wiggles. Due to the positivity, the set of equations can be solved by collective relaxation methods. A full multigrid method based on symmetric successive relaxation, full weighting, bilinear interpolation and W-cycle is presented. Typical full multigrid efficiency is achieved for the GAMM transonic bump test case since after the nested iteration and one cycle, the solution cannot be distinguished anymore from the fully converged solution.