High order positivity-preserving nodal discontinuous Galerkin methods for anisotropic diffusion problems

Abstract

In this paper, we develop second and third order accurate positivity-preserving (PP) nodal discontinuous Galerkin (DG) methods for one and two dimensional anisotropic diffusion problems. The key idea is to first represent the cell average of its numerical approximation as a weighted summation of Gaussian quadrature point values used in the updating of nodal DG methods, and then transform these Gaussian quadrature point values to some other special chosen point values. We prove that by taking parameters in the definition of nodal DG methods appropriately, together with a suitable time stability condition, the cell averages can be kept positive. A polynomial scaling limiter is then applied to obtain positive numerical approximations on the whole cell without sacrificing accuracy. Stability analysis without the PP limiter is also rigorously established. Numerical experiments are performed to demonstrate desired orders of accuracy, PP and good performances of our proposed approach.