Prof. Weizhu Bao   
Department of Mathematics, National University of Singapore, Singapore

Speech Title: Multiscale methods and analysis for the highly oscillatory nonlinear Klein-Gordon equation

Abstract

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.

Biography

Professor Weizhu Bao received his PhD degree from Tsinghua University in 1995. He was an Associate Professor at Tsinghua University, and Visisting Assistant Professor at Georgia Institute of Technology and University of Wisconsin-Madison. Currently he is a Full Professor at Department of Mathematics, National University of Singapore. Professor Bao is well known for his work in applied mathematics with applications in quantum physics, chemistry and materials science, especially Bose-Einstein condensation and highly oscillatory partial differential equations. He has received many awards, including the Beijing Science and Technology Award in 2003, and the Feng Kang Prize in Scientific Computing in 2013. Professor Bao was also an invited speaker at the International Congress of Mathematicians in 2014. He was in the editorial board of many famous journals, including SIAM Journal of Scientific Computing, Communications in Mathematical Sciences, Journal of Computational Mathematics, and so on.