期刊
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
卷 -, 期 -, 页码 -出版社
GLOBAL SCIENCE PRESS
DOI: 10.4208/aamm.OA-2022-0188
关键词
Camassa-Holm equation; quadratic auxiliary variable; high-order energy-preserving schemes; symplectic Runge-Kutta methods
资金
- Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems [202002]
- Fundamental Research Funds for the Central Universities [NS2022070]
- Natural Science Foundation of Jiangsu Province [BK20220131]
- National Natural Science Foundation of China [12171245, 12271252, 12071216, 12201297]
- Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems [202001]
- Science Challenge Project [TZ2018002]
- National Science and Technology Major Project [J2019-II-0007-0027]
- National Key Research and Development Program of China [2018YFC1504205]
In this paper, a quadratic auxiliary variable (QAV) technique is used to develop energy-preserving algorithms for the Camassa-Holm equation. The technique transforms the original equation and discretizes it to obtain a class of fully discrete schemes. The proposed methods are proven to be energy-preserving, and numerical experiments confirm their accuracy, conservative property, and efficiency.
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.
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