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

doi:10.1007/s42967-020-00100-7

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (1): 84-107.doi: 10.1007/s42967-020-00100-7

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

Hongjuan Zhang¹, Boying Wu², Xiong Meng²

1 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China;
2 School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received:2020-08-07 Revised:2020-10-23 Online:2022-03-20 Published:2022-03-01
Contact: Xiong Meng, Hongjuan Zhang, Boying Wu E-mail:xiongmeng@hit.edu.cn;18B912028@stu.hit.edu.cn;mathwby@hit.edu.cn
Supported by:
The research of Boying Wu was supported by the National Natural Science Foundation of China Grants U1637208 and 71773024. The research of Xiong Meng was supported by the National Natural Science Foundation of China Grant 11971132.

Abstract

Abstract: In this paper, we consider the local discontinuous Galerkin method with generalized alternating numerical fuxes for two-dimensional nonlinear Schrödinger equations on Cartesian meshes. The generalized fuxes 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. Numerical experiments are given to demonstrate the theoretical results.

Key words: Local discontinuous Galerkin method, Two-dimensional nonlinear Schrödinger equation, Generalized alternating fuxes, Optimal error estimates

CLC Number:

Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.

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

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