报告题目：QUADRATIC AUXILIARY VARIABLE RUNGE-KUTTA METHODS FOR THE CAMASSA-HOLM EQUATION

报 告 人：王雨顺教授（南京师范大学）

报告时间：2022年12月22日  16:00

报告地点：腾讯会议（231-581-820）

报告摘要：

In this paper, we take the Camassa-Holm equation as an example to propose a novel class of Runge-Kutta methods for conservative systems, 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 dirctly apply the symplectic RK methods to develop energy-preserving algorithms for general conservation systems with any polynomial energy.

报告人简介：

王雨顺，长期从事保结构算法及其应用研究，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，获得江苏省科技进步奖、高等学校科学研究优秀成果奖二等奖、江苏省数学成就奖。专著《偏微分方程保结构算法》获得第三届中国政府出版奖-图书奖。现任期刊International Journal of Computer Mathematics、《计算数学》编委。江苏省江苏数学会秘书长。