A Multigrid Method for Steady Euler Equations, Based on Flux-Difference Splitting with Respect to Primitive Variables

A flux-difference Splitting method for steady Euler equations, resulting in a Splitting with respect to primitive variables is introduced. This Splitting is applied to finite volumes centered around the vertices of the computational grid. The discrete set of equations is both conservative and positive.Due to the positivity, the solution can be obtained by collective variants of relaxation methods, that can be brought into multigrid form. Two full multigrid methods are presented. As restriction operator, both use full weighting within the flow field and injection at the boundaries. Bilinear interpellation is used as prolongation. The cycle is of V7-type. The first method uses symmetric successive underrelaxation, while the second uses Jacobi-iteration. In terms of cycles, the successive formulation is the most efficient while in terms of computer time, due to the vectorizability the second formulation is most efficient.Due to the conservativity, the algebraically exact flux-difference splitting and the positivity, the solution for transonic flow shows shocks represented as sharp discontinuities, without wiggles.