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Volume 15, Issue 5
Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach

Yuezheng Gong, Qi Hong, Chunwu Wang & Yushun Wang

Adv. Appl. Math. Mech., 15 (2023), pp. 1233-1255.

Published online: 2023-06

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  • Abstract

In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

  • AMS Subject Headings

65M06, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-15-1233, author = {Gong , YuezhengHong , QiWang , Chunwu and Wang , Yushun}, title = {Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2023}, volume = {15}, number = {5}, pages = {1233--1255}, abstract = {

In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2022-0188}, url = {http://global-sci.org/intro/article_detail/aamm/21775.html} }
TY - JOUR T1 - Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach AU - Gong , Yuezheng AU - Hong , Qi AU - Wang , Chunwu AU - Wang , Yushun JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1233 EP - 1255 PY - 2023 DA - 2023/06 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2022-0188 UR - https://global-sci.org/intro/article_detail/aamm/21775.html KW - Camassa-Holm equation, quadratic auxiliary variable, high-order energy-preserving schemes, symplectic Runge-Kutta methods. AB -

In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

Yuezheng Gong, Qi Hong, Chunwu Wang & Yushun Wang. (2023). Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach. Advances in Applied Mathematics and Mechanics. 15 (5). 1233-1255. doi:10.4208/aamm.OA-2022-0188
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