Title: Multiscale methods and analysis for the Dirac equation

in the nonrelativistic limit regime

Speaker: Professor Weizhu Bao

       Department of Mathematics

       National University of Singapore

Datetime: 11:00a.m.-12:00a.m. March 5th

Venus: A1-407 Main Building

Abstract

In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit 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 and indefinite, 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 the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit 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 Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime.

This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.

[1] W. Bao, Y. Cai, X. Jia and Q. Tang, Numerical methods and comparison for the Dirac

equation in the nonrelativistic limit regime, J. Sci. Comput., 71 (2017), pp. 1094-1134. 

[2] W. Bao, Y. Cai, X. Jia and Q. Tang, A uniformly accurate multiscale time integrator

pseudospectral method for the Dirac equation in the nonrelativistic limit regime,

SIAM J. Numer. Anal., 54 (2016), pp. 1785-1812.

[3] W. Bao, Y. Cai, X. Jia and J. Yin, Error estimates of numerical methods for the nonlinear

Dirac equation in the nonrelativistic limit regime, Sci. China Math., 59 (2016), pp. 1461-1494.

  Professor Weizhu BAO is currently 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 held the Provost's Chair Professorship at Department of  Mathematics, 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 was 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 conference including the Invited Speaker at the International Congress of Mathematicians (ICM) in 2014.