Volume 16, Issue 2
Numerical Analysis of an Energy-Conservation Scheme for Two-Dimensional Hamiltonian Wave Equations with Neumann Boundary Conditions

Changying Liu, Wei ShiXinyuan Wu

Int. J. Numer. Anal. Mod., 16 (2019), pp. 319-339.

Published online: 2018-10

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  • 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 Hamlitonian 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.

  • Keywords

Two-dimensional Hamiltonian wave equation finite difference method Neumann boundary conditions energy-conservation algorithm average vector field formula.

  • AMS Subject Headings

35C15 65M06 65M12 65M20 65M70.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

chyliu88@gmail.com (Changying Liu)

wilburs@illinois.edu (Wei Shi)

xywu@nju.edu.cn (Xinyuan Wu)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-16-319, author = {Liu , Changying and Shi , Wei and Wu , Xinyuan}, title = {Numerical Analysis of an Energy-Conservation Scheme for Two-Dimensional Hamiltonian Wave Equations with Neumann Boundary Conditions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {16}, number = {2}, pages = {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 Hamlitonian 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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12806.html} }
TY - JOUR T1 - Numerical Analysis of an Energy-Conservation Scheme for Two-Dimensional Hamiltonian Wave Equations with Neumann Boundary Conditions AU - Liu , Changying AU - Shi , Wei AU - Wu , Xinyuan JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 319 EP - 339 PY - 2018 DA - 2018/10 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12806.html KW - Two-dimensional Hamiltonian wave equation KW - finite difference method KW - Neumann boundary conditions KW - energy-conservation algorithm KW - average vector field formula. AB -

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 Hamlitonian 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.

Changying Liu, Wei Shi & Xinyuan Wu. (2020). Numerical Analysis of an Energy-Conservation Scheme for Two-Dimensional Hamiltonian Wave Equations with Neumann Boundary Conditions. International Journal of Numerical Analysis and Modeling. 16 (2). 319-339. doi:
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