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Volume 35, Issue 6
A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs

Kai Liu, Xinyuan Wu & Wei Shi

DOI: 10.4208/jcm.1612-m2016-0604

J. Comp. Math., 35 (2017), pp. 780-800.

Published online: 2017-12

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  • Abstract

The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative (dissipative) nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.

  • Keywords

Conservative (dissipative) wave PDEs, Structure-preserving algorithm, Linearly-fitted, Average Vector Field formula, Sine-Gordon equation.

  • AMS Subject Headings

65L05, 65L07, 65L20, 65P10, 34C15.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

laukai520@163.com (Kai Liu)

xywu@nju.edu.cn (Xinyuan Wu)

shuier628@163.com (Wei Shi)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-780, author = {Liu , KaiWu , Xinyuan and Shi , Wei}, title = {A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {6}, pages = {780--800}, abstract = {

The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative (dissipative) nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1612-m2016-0604}, url = {http://global-sci.org/intro/article_detail/jcm/10494.html} }
TY - JOUR T1 - A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs AU - Liu , Kai AU - Wu , Xinyuan AU - Shi , Wei JO - Journal of Computational Mathematics VL - 6 SP - 780 EP - 800 PY - 2017 DA - 2017/12 SN - 35 DO - http://doi.org/10.4208/jcm.1612-m2016-0604 UR - https://global-sci.org/intro/article_detail/jcm/10494.html KW - Conservative (dissipative) wave PDEs, Structure-preserving algorithm, Linearly-fitted, Average Vector Field formula, Sine-Gordon equation. AB -

The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative (dissipative) nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.

Kai Liu, Xinyuan Wu & Wei Shi. (2020). A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs. Journal of Computational Mathematics. 35 (6). 780-800. doi:10.4208/jcm.1612-m2016-0604
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