Invited Lecture
Stability of IMEX-LDG schemes for linearized KdV equations
Division of Applied Mathematics, Brown University, USA
Abstract
Stable semi-discrete local discontinuous Galerkin methods with optimal error estimates for solving the linearized KdV equations were obtained in the literature some years ago. Explicit Runge-Kutta methods can be used to discretize the time variable, however this would lead to a severe time step restriction $\Delta t = O(\Delta x^3)$. IMEX methods, which treat the dispersive third order spatial derivative term implicitly and the convective first order spatial derivative term explicitly, could be expected to reach a convective time step restriction $\Delta t = O(\Delta x)$ for stability. In this talk, we report our recent work in proving the somewhat surprising result, that the IMEX-LDG schemes for solving the linearized KdV equations, with up to second order accuracy in time, could be designed to have an unconditional stability, in the sense that stability can be proved for $\Delta t <= au$ where the constant $ au$ is independent of the spatial mesh size $\Delta x$. Optimal $ L^2$ error estimates are also proved.
This is a joint work with Haijin Wang, Qi Tao and Qiang Zhang.