报告题目:A class of efficient Hamiltonian conservative spectral methods for Korteweg-de Vries equations

报告人：曹外香 副教授（北京师范大学）

报告时间：2023年6月20日（周二）晚上19:00-20:00

报告地点：#腾讯会议：742-566-559

Abstract: 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.

About the Speaker: 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. Sci. Comput. 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。