A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations

In this paper, we consider the local discontinuous Galerkin method with generalized alternating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Cartesian meshes. The generalized fluxes not only lead to a smaller magnitude of the errors, but can guarantee an energy conservative property that is useful for long time simulations in resolving waves. By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators, two energy equations are established and stability results containing energy conservation of the prime variable as well as auxiliary variables are shown. To derive optimal error estimates for nonlinear Schrödinger equations, an additional energy equation is constructed and two a priori error assumptions are used. This, together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition, implies optimal order of k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document}. Numerical experiments are given to demonstrate the theoretical results.