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Advanced Numerical Methods in the Mathematical Sciences

Chi-Wang Shu, Brown University
IMEX Time Marching for Discontinuous Galerkin Methods


  • Chi-Wang Shu
  • Haijin Wang
  • Qiang Zhang
  • Yunxian Liu


For discontinuous Galerkin methods approximating convection diffusion equations, explicit time marching is expensive since the time step is restricted by the square of the spatial mesh size. Implicit methods, however, would require the solution of non-symmetric, non-positive definite and nonlinear systems, which could be difficult. The high order accurate implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the diffusion term implicitly (often linear, resulting in a linear positive-definite solver) and the convection term (often nonlinear) explicitly, can greatly improve computational efficiency. We prove that certain IMEX time discretizations, up to third-order accuracy, coupled with the local discontinuous Galerkin method for the diffusion term treated implicitly, and the regular discontinuous Galerkin method for the convection term treated explicitly, are unconditionally stable (the time step is upper-bounded only by a constant depending on the diffusion coefficient but not on the spatial mesh size) and optimally convergent. The results also hold for the drift-diffusion model in semiconductor device simulations, where a convection diffusion equation is coupled with an electrical potential equation. Numerical experiments confirm the good performance of such schemes. This is a joint work with Haijin Wang, Qiang Zhang and Yunxian Liu.