报告题目：Multiscale methods and analysis for the highly oscillatory nonlinear Klein-Gordon equation

报 告 人：包维柱教授（National University of Singapore）

报告时间：2022年12月1日  14:30

报告地点：腾讯会议（672801482）

报告摘要：

In this talk, I begin with the nonlinear Klein-Gordon equation (NKGE) under two important parameter regimes, i.e. one is nonrelativisitic regime and the other is long-time dynamics with weak nonlinearity or small initial data, while the NKGE is highly oscillatory. I first review our recent works on numerical methods and analysis for solving the NKGE in 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 Schrodinger equation perturbed with a wave operator. Based on a large-small amplitude wave decompostion to the solution of the NKGE, a multiscale method 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. Finally, I discuss issues related error bounds of different numerical methods for the long-time dynamics of NKGE with weak nonlinearity and applications to several highly oscillatory dispersive partial differential equations.

报告人简介：

Professor Weizhu BAO is a Professor at Department of Mathematics, National University of Singapore (NUS). He got his PhD from Tsinghua Univeristy in 1995 and afterwards he had postdoc and faculty positions at Tsinghua University, Imperial College in UK, Georgia Institute of Technology and University of Wisconsin at Madison in USA. He joined NUS as an Assistant Professor in 2000 and was promoted to Professor in 2009. He had been appointed as the Provost's Chair Professorship at NUS during 2013 -- 2016. His research interests include numerical methods for partial differential equations, scientific computing/ numerical analysis, analysis and computation for problems from physics, chemistry, biology and engineering sciences. He has made significant contributions in modeling and simulation of Bose-Einstein condensation, solid-state dewetting; and in multiscale methods and analysis for highly oscillatory PDEs. He had been on the Editorial Board of SIAM Jourmal on Scientific Computing during 2009- -2014 and is currently on the Editorial Board of SIAM Joumal of Numerical Analysis. He was awarded the Feng Kang Prize in Scientific Computing by the Chinese Computational Mathematics Society in 2013. He has been invited to give plenary and/or invited talks in many interational conferences including an Invited Speaker at the International Congress of Mathematicians (ICM) in 2014. He was elected a Fellow of the American Mathematical Society (AMS Fellow), a Fellow of the Society of Industrial and Applied Mathematics (SIAM Fellow) and a Fellow of Singapore National Academy of Science (SNAS Fellow) in 2022.