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A semi-Lagrangian discontinuous Galerkin (DG) -- local DG method for solving convection-diffusion equations
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 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.Comment: 34 pages; 16 figures; Journal of Computational Physics, accepte