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In this talk we give a short summary of our recent work [5], jointly with Z. Sun, on establishing the equivalency of the Runge-Kutta discontinuous Galerkin (RKDG) methods and a class of Lax-Wendroff discontinuous Galerkin (LWDG) methods for solving linear conservation laws, as well as on stability analysis and error estimates for the LWDG methods for solving one- and two-dimensional linear conservation laws, regardless of whether they are equivalent to the RKDG methods or not. Our stability analysis includes multidimensional problems with divergence-free coefficients, and our error estimates include those for both the solution u and its first order time derivative ut. [ABSTRACT FROM AUTHOR]

In this talk we give a short summary of our recent work [9, 10, 11, 7], jointly with H. Wang, Q. Zhang, S. Wang and Y. Liu, on the development, analysis and application of implicit-explicit (IMEX) Runge-Kutta or multi-step time marching methods for discontinuous Galerkin (DG) methods approximating convection diffusion equations. For such DG methods, explicit time marching is expensive since the time step is restricted by the square of the spatial mesh size, while fully implicit methods would require the solution of a non-symmetric, non-positive definite and possibly nonlinear system in each time step. The high order accurate IMEX Runge-Kutta or multi-step time marching would treat the diffusion term implicitly (which is often linear, resulting in a linear positive-definite solver) and the convection term (often nonlinear) explicitly, hence it can greatly improve computational efficiency. We prove that certain IMEX time discretizations, up to third order accuracy, coupled with local DG (LDG) method for the diffusion term treated implicitly, and regular DG 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 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. [ABSTRACT FROM AUTHOR]