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

来源: 04-06

时间:Thur.,16:00-17:00, April 6th, 2023

地点:Lecture Hall, Floor 3 Jin Chun Yuan West Building

组织者:应用与计算数学团队

主讲人:Waixiang Cao 曹外香 Beijing Normal University

Speaker

曹外香,北京师范大学数学科学学院副教授,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., 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青年基金一项,面上项目两项。


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.

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