A reconstructed discontinuous approximation method for the porous medium equation

The porous medium equation (PME) describes the fluid flow in saturates porous medium. In this paper, we develop an interior penalty discontinuous Galerkin method in a reconstructed discontinuous approximation (RDA) space (called as RDA method) for solving the porous medium equation. The core of the RDA method is the utility of a reconstructed space with only one degree of freedom per element, which is derived by solving the least squares fitting problem on each element patch. The RDA method inherits the advantages of the discontinuous Galerkin (DG) method since the RDA space is a subspace of the DG space. In addition, the freedom of the arbitrary mth (m ≥1) order RDA method is equal to the number of mesh elements nₑ, which is much less than that of the DG method (dim(Pᵐ)nₑ). An explicit-implicit-null discretization is used in time, and a high-order post-processing of the solution is used to eliminate the spurious oscillations at the free boundary. We theoretically prove that the method has high-order accuracy for the smooth solution. Numerical experiments of one- and two-dimensional porous medium equations are presented to validate the theoretical analysis and illustrate the performance of the proposed method.