A semi-Lagrangian discontinuous Galerkin (DG) -- local DG method for solving convection-diffusion-reaction equations
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In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion-reaction equations. The method generalizes our previous work on developing the SLDG method for transport equations (J. Sci. Comput. 73: 514-542, 2017), making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion are discretized by local DG (LDG) method and are integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the `SLDG-LDG' method, which enjoy many attractive features of the DG and SL methods. These include the uniformly high order accuracy (e.g. third order) in space and in time, compact, mass conservative, and stability under large time stepping size. An L^2 stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions.
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