A class of efficient Hamiltonian conservative spectral methods for Korteweg-de Vries equations

主讲人：曹外香 北京师范大学副教授

时间：2023年10月20日14：00

地点：腾讯会议 478 162 944

举办单位：数理学院

主讲人介绍：曹外香，北京师范大学数学科学学院副教授，美国布朗大学访问学者，研究方向为偏微分方程数值解法和数值分析，主要研究有限元方法、有限体积方法，间断有限元方法高效高精度数值计算。主要结果发表在SIAM J. Numer. Anal., Math. Comp., J. Comput.Phys. 等期刊上。曾获中国博士后基金一等资助和特别资助，广东省自然科学二等奖，主持国家自然科学基金面上项目、国家自然科学基金青年基金等项目。

内容介绍：In this talk, we present and introduce two efficient Hamiltonian conservative fully discrete numerical schemes for Korteweg-de Vries equations. The new numerical schemes are constructed by using time-stepping spectral Petrov-Galerkin (SPG) or Gauss collocation (SGC) methods for the temporal discretization coupled with the $p$-version/spectral local discontinuous Galerkin (LDG) methods for the space discretization. We prove that the fully discrete SPG-LDG scheme preserves both the momentum and the Hamilton energy exactly for generalized KdV equations. While the fully discrete SGC-LDG formulation preserves the momentum and the Hamilton energy exactly for linearized KdV equations. As for nonlinear KdV equations, the SGC-LDG scheme preserves the momentum exactly and is Hamiltonian conserving up to some spectral accuracy. Furthermore, we show that the semi-discrete $p$-version LDG methods converge exponentially with respect to the polynomial degree. The numerical experiments are provided to demonstrate that the proposed numerical methods preserve the momentum, $L^2$ energy and Hamilton energy and maintain the shape of the solution phase efficiently over long time period.