Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach  

作　　者：Yuezheng Gong Qi Hong Chunwu Wang Yushun Wang 

机构地区：[1]School of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing,Jiangsu 211106,China [2]Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA),MIIT,Nanjing,Jiangsu 211106,China [3]Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems,School of Mathematical Sciences,Nanjing Normal University,Nanjing,Jiangsu 210023,China

出　　处：《Advances in Applied Mathematics and Mechanics》2023年第5期1233-1255,共23页应用数学与力学进展（英文）

基　　金：Yuezheng Gong’s work is partially supported by the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202002);the Fundamental Research Funds for the Central Universities(Grant No.NS2022070);the Natural Science Foundation of Jiangsu Province(Grant No.BK20220131);the National Natural Science Foundation of China(Grants Nos.12271252 and 12071216);Qi Hong’s work is partially supported by the National Natural Science Foundation of China(Grants No.12201297);the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202001);Chunwu Wang’s work is partially supported by Science Challenge Project(Grant No.TZ2018002);National Science and Technology Major Project(J2019-II-0007-0027);Yushun Wang’s work is partially supported by the National Key Research and Development Program of China(Grant No.2018YFC1504205);the National Natural Science Foundation of China(Grants No.12171245).

摘　　要：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.

关 键 词：Camassa-Holm equation quadratic auxiliary variable high-order energy-preserving schemes symplectic Runge-Kutta methods 

分 类 号：O17[理学—数学]