Adaptive local discontinuous Galerkin approximation to Richards’ equation

We propose a spatially and temporally adaptive solution to Richards’ equation based upon a local discontinuous Galerkin approximation in space and a high-order, backward difference method in time. We cast our approach in terms of a general, decoupled adaption algorithm based upon operators. We define non-unique instances of all operators to result in an adaption method from within the general class of methods that is defined. We formally decouple the spatial adaption from the temporal adaption using a method of lines approach and limit the temporal truncation error so that the total error is dominated by the spatial component. We use a multiple grid approach to guide adaption and support the data structures. Spatial adaption decisions are based upon error and regularity indicators, which are economical to compute. The resultant methods are compared for two test problems. The results show that the proposed adaption methods are superior to methods that adapt only in time and that in cases in which the problem has sufficient smoothness, adapting the order of the elements in addition to the grid spacing can further improve the efficiency of this robust solution approach.