QUADRATIC AUXILIARY VARIABLE RUNGE-KUTTA METHODS FOR THE CAMASSA-HOLM EQUATION

主讲人：王雨顺 南京师范大学教授

时间：2022年5月25日10：00

地点：腾讯会议 945 830 592

举办单位：数理学院

主讲人介绍：王雨顺，南京师范大学教授、博导。长期从事保结构算法及其应用研究，已经主持7项国家基金委项目，同时作为主要成员参加科技部“863”课题、“973”项目、“863”计划、基金委重点项目，入选江苏省“333”工程、青蓝工程学术带头人、江苏省“六大人才高峰”高层次人才；江苏省创新团队主持人；获得江苏省科学技术奖，江苏省数学成就奖。专著《偏微分方程保结构算法》获得中国政府图书奖。现任期刊International Journal of Computer Mathematics、《计算数学》编委，江苏省计算数学分会秘书长。

内容介绍：In this paper, we propose a novel class of Runge-Kutta methods for the Camassa-Holm equation, which is named quadratic auxiliary variable Runge-Kutta (QAVRK) methods. We first introduce an auxiliary variable that satisfies a quadratic equation and rewrite the original energy into a quadratic functional. With the aid of the energy variational principle, the original system is then reformulated into an equivalent form with two strong quadratic invariants, where one is induced by the quadratic auxiliary variable and the other is the modified energy. Starting from the equivalent model, we employ RK methods satisfying the symplectic condition for time discretization, which naturally conserve all strong quadratic invariants of the new system. The resulting methods are shown to inherit the relationship between the auxiliary variable and the original one, and thus can be simplified by eliminating the auxiliary variable, which leads to a new class of QAVRK schemes. Furthermore, the QAVRK methods are proved rigorously to preserve the original energy conservation law. Numerical examples are presented to confirm the expected order of accuracy, conservative property and efficiency of the proposed schemes. This numerical strategy makes it possible to directly apply the symplectic RK methods to develop energy-preserving algorithms for general conservation systems with more than polynomial energy.