Maximum-principle-preserving third-order local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes

Abstract

Local discontinuous Galerkin (LDG) methods are popular for convection-diffusion equations. In LDG methods, we introduce an auxiliary variable p to represent the derivative of the primary variable u, and solve them on the same mesh. It is well known that the maximum-principle-preserving (MPP) LDG method is only available up to second-order accuracy. Recently, we introduced a new algorithm, and solve u and p on different meshes, and obtained stability and optimal error estimates. In this paper, we will continue this approach and construct MPP third-order LDG methods for convection-diffusion equations on overlapping meshes. The new algorithm is more flexible and does not increase any computational cost. Numerical evidence will be given to demonstrate the accuracy and good performance of the third-order MPP LDG method.