The Direct Discontinuous Galerkin Method For Richards' Equation

We consider the numerical approximation to Richards'equation (model for flow in porous media) because of its hydrological significance and intrinsic merit as nonlinear advection-diffusion model that admits sharp fronts in space and time that pose a great challenge to conventional methods. In [12][13], the author introduced direct discontinuous Galerkin (DDG) method to advection-diffusion equations. In this paper we try to apply DDG method to Richards'equation. Because of its nonlinearity and hyperbolicity, rigorous time-step is necessary for explicit scheme in time, that is fatal to long-time simulation. To overcome this difficulty we combine backward difference method for time integration with spatial discretization approach based upon DDG method. In the direct weak formulation of Richards'equation we linearize the nonlinear part. Then we get a set of linear equations, whose coefficient matrix is sparse and diagonally dominant. It can be easily solved. Error estimate is given for fully discrete scheme in this paper. Applying DDG method to a set of problems we verify the conclusion. Then numerical experiments are carried out for Richards'equation to show the robustness and efficiency of the method. Finally we give a higher-order formulation for temporal approximation.