A Note on the Use of Symmetric Line Gauss–Seidel for the Steady Upwind Differenced Euler Equations

Symmetric Line Gauss–Seidel (SLGS) relaxation, when used to compute steady solutions to the upwind differenced Euler equations of gas dynamics, is shown to be unstable. The instability occurs for the long waves. If SLGS is used in a multigrid scheme, stability is restored. However, the use of an unstable relaxation scheme will not provide a robust multigrid code. Damped Symmetric Point Gauss–Seidel relaxation is stable and provides similar multigrid convergence rates at much lower cost. However, it fails if the flow is aligned with the grid over a substantial part of the computational domain. Damped Alternating Direction Line Jacobi relaxation can overcome this problem.