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

## TL;DR

This paper introduces a high-order semi-Lagrangian discontinuous Galerkin method for efficiently solving linear convection-diffusion equations, combining characteristics evolution with local DG discretization for diffusion.

## Contribution

It extends previous semi-Lagrangian DG methods to handle diffusion and source terms, providing high-order accuracy, stability, and efficiency for convection-diffusion problems.

## Key findings

- Achieves high-order accuracy in space and time.
- Demonstrates stability under large time steps.
- Shows effectiveness through numerical tests in 1D and 2D.

## Abstract

In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion 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 term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the `SLDG-LDG' method and enjoys 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.