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

报告时间：2023年3月22日 10:00-11:30

报告地点：数学楼 2-3会议室

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. This is based on joint works with Yongyong Cai, Xuchun Dong, Yue Feng, Chunmei Su, Wenfan Yi and Xiaofei Zhao.

个人简介：Professor Weizhu BAO is currently a Professor at Department of Mathematics and Vice Dean for Graduate Studies and Academic Affair of Faculty of Science (FoS), National University of Singapore (NUS). 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 had been on the Editorial Board of SIAM Journal on Scientific Computing during 2009--2014. 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 international conferences including the Invited Speaker at the International Congress of Mathematicians (ICM) in 2014.