Assessment of implicit operators for the upwind point Gauss–Seidel method on unstructured meshes

The effect of the numerical dissipation level of implicit operators on the stability and convergence characteristics of the upwind point Gauss–Seidel (GS) method for solving the Euler equations was studied through the von Neumann stability analysis and numerical experiments. The stability analysis for linear model equations showed that the point GS method is unstable even for very small CFL numbers when the numerical dissipation level of the implicit operator is equivalent to that of the explicit operator. The stability restriction is rapidly alleviated as the dissipation level of the implicit operator increases. The instability predicted by the linear stability analysis was further amplified as the flow problems became stiffer due to the presence of the shock wave or the refinement of the mesh. It was found that for the efficiency and the robustness of the upwind point GS method, the numerical flux of the implicit operator needs to be more dissipative than that of the explicit operator.