Local Discontinuous Galerkin Methods with Multistep Implicit-Explicit Time Discretization for Nonlinear Schrödinger Equations

In this paper, we investigate the local discontinuous Galerkin (LDG) methods coupled with multistep implicit-explicit (IMEX) time discretization to solve one-dimensional and two-dimensional nonlinear Schr & ouml;dinger equations. In this approach, the nonlinear terms are treated explicitly, while the linear terms are handled implicitly. By the skew symmetry property of LDG operators and the properties of Gauss-Radau projections, we obtain error estimates for the prime and auxiliary variables, as well as the estimate for the time difference of the prime variables. These results, together with a carefully chosen numerical initial condition, allow us to obtain the optimal error estimate in both space and time for the fully discrete scheme. Numerical experiments are performed to verify the accuracy and efficiency of the proposed methods.