A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations

For constructing high order accurate positivity-preserving schemes for convection-diffusion equations, we construct a simple positivity-preserving diffusion flux. Discontinuous Galerkin (DG) schemes with such a positivity-preserving diffusion flux are nonlinear schemes, which can be regarded as a reduction of the high order positivity-preserving DG schemes for compressible Navier-Stokes equations in [1] to scalar diffusion operators. In this paper we focus on the local DG method to discuss how to apply such a flux. A limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity-preserving in the sense that DG schemes with this flux satisfies a weak positivity property. Together with a positivity-preserving limiter, high order DG schemes with strong stability preserving time discretizations can be rendered positivity-preserving without losing conservation or high order accuracy for convection-diffusion problems with periodic boundary conditions or a special class of Dirichlet or Neumann boundary conditions. Numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme.