Multigrid methods for high-order accurate fully implicit simulation of flowin porous media

High-order accurate finite difference schemes are widely used to avoid the detrimental effects of numerical diffusion in first-order upwind schemes. If an implicit time integration scheme is employed, we have to solve large systems of nonlinear equations in every time step. The price paid for the high-order accuracy is a larger discretization stencil. In this paper we consider the use of multigrid methods for the iterative solutions of these systems of equations. We consider both a direct multigrid approach and a defect correction approach, in which only first-order accurate discretized problems have to be solved. In both approaches we do not need to store the Jacobean matrix. Therefore the memory requirements are moderate, and very fine grid simulations are feasible on a standard workstation.