Multiscale methods and analysis for the highly oscillatory nonlinear Klein-Gordon equation
数学公众报告(120周年校庆系列第77场)
报告题目(Title)：Multiscale methods and analysis for the highly oscillatory nonlinear Klein-Gordon equation
报告人(Speaker)：Professor Weizhu Bao (National University of Singapore)
地点(Place)：后主楼1124
时间(Time)：2023年3月17日(周五), 10:30-11:30
邀请人(Inviter)：蔡勇勇
报告摘要
In this talk, I will review our recent works on numerical methods and analysis for solving the highly oscillatory nonlinear Klein-Gordon equation (NKGE) including the nonrelativistic regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the NKGE. Rigorious error estimates show that the EWI spectral method show much better temporal resolution than the FDTD methods for the NKGE in the nonrelativistic regime. In order to design a multiscale method for the NKGE, we establish error estimates of FDTD and EWI spectral methods for the nonlinear Schroedinger equation perturbed with a wave operator. Based on a large-small amplitude wave decompostion to the solution of the NKGE, a multiscale time integrator (MTI) is presented for discretizing the NKGE in the nonrelativistic regime. Rigorous error estimates show that this multiscale method converges uniformly in spatial/temporal discretization with respect to the small parameter for the NKGE in the nonrelativistic regime. Extension to the long-time dynamics of the NKGE with weak nonlinearity is discussed and improved uniform error bounds on time-splitting spectral method are presented based on a new technique -- regularity compensation oscillation. Finally, applications to several high oscillatory dispersive partial differential equations will be discussed.
主讲人简介
包维柱教授现为新加坡国立大学数学系教授，理学院副院长，新加坡科学院院士,SIAM Fellow, AMS Fellow，博士毕业于清华大学，曾获冯康科学计算奖，应邀在韩国举行的第26届国际数学家大会上作45分钟邀请报告。包维柱教授长期从事科学与工程计算研究，主要工作涉及偏微分方程数值方法及其在量子物理、流体和材料中的应用。
