Numerical Analysis of an Energy-Conservation Scheme for Two-Dimensional Hamiltonian Wave Equations with Neumann Boundary Conditions
Year: 2019
Author: Changying Liu, Wei Shi, Xinyuan Wu
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 2 : pp. 319–339
Abstract
In this paper, an energy-conservation scheme is derived and analysed for solving Hamiltonian wave equations subject to Neumann boundary conditions in two dimensions. The energy-conservation scheme is based on the blend of spatial discretisation by a fourth-order finite difference method and time integration by the Average Vector Field (AVF) approach. The spatial discretisation via the fourth-order finite difference leads to a particular Hamiltonian system of second-order ordinary differential equations. The conservative law of the discrete energy is established, and the stability and convergence of the semi-discrete scheme are analysed. For the time discretisation, the corresponding AVF formula is derived and applied to the particular Hamiltonian ODEs to yield an efficient energy-conservation scheme. The numerical simulation is implemented for various cases including a linear wave equation and two nonlinear sine-Gordon equations. The numerical results demonstrate the spatial accuracy and the remarkable energy-conservation behaviour of the proposed energy-conservation scheme in this paper.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2019-IJNAM-12806
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 2 : pp. 319–339
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Two-dimensional Hamiltonian wave equation finite difference method Neumann boundary conditions energy-conservation algorithm average vector field formula.