Analysis of Local Discontinuous Galerkin Methods with Implicit-Explicit Time Marching for Linearized KdV Equations

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2222-2248, October 2024.
Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEX-LDG schemes, combining local discontinuous Galerkin spatial discretization with implicit-explicit Runge–Kutta temporal discretization, for the linearized one-dimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes.