High-order accurate and fully implicit finite difference schemes are widely used for multiphase flow problems. In this paper we analyze the use of point Gauss-Seidel relaxation in a nonlinear multigrid method for the resulting nonlinear systems of equations. Point Gauss-Seidel is unstable for calculating the steady-state of high-order accurate discretizations of the 2D convection equation. Here we present a local Fourier mode smoothing analysis for the transient case. It appears that point Gauss-Seidel is a good smoother provided that the time step is taken small enough. Numerical computations show good multigrid convergence rates for typical test problems.

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Molenaar, G. (1997). Nonlinear multigrid for fully-implicit and high-order accurate simluation of multiphase flow in porous media. CWI.