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In this paper, we propose an efficient high order semi-Lagrangian (SL)discontinuous Galerkin (DG) method for solving linear convection-diffusionequations. The method generalizes our previous work on developing the SLDGmethod for transport equations (J. Sci. Comput. 73: 514-542, 2017), making itcapable of handling additional diffusion and source terms. Within the DGframework, the solution is evolved along the characteristics; while thediffusion term is discretized by the local DG (LDG) method and integrated alongcharacteristics by implicit Runge-Kutta methods together with source terms. Theproposed method is named the `SLDG-LDG' method and enjoys many attractivefeatures of the DG and SL methods. These include the uniformly high orderaccuracy (e.g. third order) in space and in time, compact, mass conservative,and stability under large time stepping size. An $L^2$ stability analysis isprovided when the method is coupled with the first order backward Eulerdiscretization. Effectiveness of the method are demonstrated by a group ofnumerical tests in one and two dimensions.

A semi-Lagrangian discontinuous Galerkin (DG) -- local DG method for solving convection-diffusion equations