A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation

JOURNAL OF COMPUTATIONAL PHYSICS

卷:321

Article

Liu, Hailiang¹  Yi, Nianyu² 

[1] Iowa State Univ, Dept Math, Ames, IA 50011 USA

[2] Xiangtan Univ, Sch Math & Computat Sci, Xiangtan 411105, Peoples R China

关键词: Discontinuous Galerkin method;    Korteweg-de Vries equation;    Conservation;    Stability;   

DOI  :  10.1016/j.jcp.2016.06.010

来源: Elsevier

【 摘 要 】

The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude errors after long time simulation. In this paper, we design, analyze and numerically validate a Hamiltonian preserving discontinuous Galerkin method for solving the Korteweg-de Vries (KdV) equation. For the generalized KdV equation, the semi-discrete formulation is shown to preserve both the first and the third conserved integrals, and approximately preserve the second conserved integral; for the linearized KdV equation, all the first three conserved integrals are preserved, and optimal error estimates are obtained for polynomials of even degree. The preservation properties are also maintained by the fully discrete DG scheme. Our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized KdV equation. We also show that the shape of the solution, after long time simulation, is well preserved due to the Hamiltonian preserving property. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free