松弛隐显Runge-Kutta方法及其在高振荡Hamilton系统的应用  

Relaxation implicit-explicit Runge-Kutta method and its applications in highly oscillatory Hamiltonian systems

作　　者：谷伟[1] 李东方 李晓西[2] 张智民 Wei Gu;Dongfang Li;Xiaoxi Li;Zhimin Zhang

机构地区：[1]中南财经政法大学统计与数学学院,武汉430073 [2]华中科技大学数学与统计学院,武汉430074 [3]华中科技大学工程建模与科学计算湖北省重点实验室,武汉430074 [4]Department of Mathematics,Wayne State University,Detroit,MI 48202,USA

出　　处：《中国科学:数学》2025年第4期829-848,共20页Scientia Sinica:Mathematica

基　　金：国家自然科学基金(批准号:11771162,12071020和12231003)资助项目。

摘　　要：保结构算法在高振荡Hamilton系统的长时间数值模拟中有着极为重要的应用.本文借助松弛技巧提出了一类用于求解高振荡Hamilton系统的松弛隐显Runge-Kutta方法.与标准隐显Runge-Kutta方法不同,该算法所具有的保结构特性使其能够应用于系统的长时间模拟.另外,该算法是线性隐式格式且能够达到任意高阶精度,从而能够极大提高系统模拟的计算效率.最后,通过多个数值算例验证本文的理论结果.Structure-preserving numerical methods have extremely important applications in long-time simulations of highly oscillatory Hamiltonian systems.In this paper,we utilize the relaxation technique and propose a family of relaxation implicit-explicit Runge-Kutta methods for solving highly oscillatory Hamiltonian systems.Contrary to the standard implicit-explicit Runge-Kutta methods,the structure-preserving properties of the proposed methods enable them to be applied to long-time simulations.Besides,the proposed methods are linearly implicit and arbitrarily high-order accurate,which can greatly improve the computational efficiency in simulations.Finally,several numerical experiments are performed to verify the theoretical results in the article.

关 键 词：高振荡Hamilton系统 松弛技巧 保结构算法 线性隐式格式 任意高阶精度 

分 类 号：O241.6[理学—计算数学]