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

主講人：曹外香 北京師范大學(xué)副教授

時間：2023年10月20日14：00

地點：騰訊會議 478 162 944

舉辦單位：數(shù)理學(xué)院

主講人介紹：曹外香，北京師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院副教授，美國布朗大學(xué)訪問學(xué)者，研究方向為偏微分方程數(shù)值解法和數(shù)值分析，主要研究有限元方法、有限體積方法，間斷有限元方法高效高精度數(shù)值計算。主要結(jié)果發(fā)表在SIAM J. Numer. Anal., Math. Comp., J. Comput.Phys. 等期刊上。曾獲中國博士后基金一等資助和特別資助，廣東省自然科學(xué)二等獎，主持國家自然科學(xué)基金面上項目、國家自然科學(xué)基金青年基金等項目。

內(nèi)容介紹：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.