报告地点：行健楼学术活动室526
Abstract: In this talk, I will review our recent works on multiscale methods and analysis for solving the highly oscillatory (nonlinear) Dirac equation 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 and indefinite, which brings significant difficulty in analysis and heavy burden in numerical computation. Rigorous error bounds are obtained for finite difference time domain (FDTD) methods, time splitting Fourier pseudospectral (TSFP) method and exponential wave integrator Fourier pseudospectral (EWI-FP), which depend explicitly on the mesh size, time step and the small parameter. Then based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and prove its error bound which uniformly accurate in term of the small dimensionless parameter. Finally, by introducing the regularity compensation oscillatory (RCO) technique, we establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the Dirac equation with small electromagnetic potentials and the nonlinear Dirac equation with weak nonlinearity. Numerical results demonstrate that our error estimates are sharp and optimal. This is a joint work with Yongyong Cai, Yue Feng, Xiaowei Jia, Qinglin Tang and Jia Yin.
报告人简介：包维柱，新加坡国立大学教务长讲席教授, 理学院副院长，SIAM Fellow，AMS Fellow，新加坡科学院院士，本科、硕士和博士毕业于清华大学，曾获冯康科学计算奖，应邀第27届国际数学家大会上作45分钟邀请报告。包教授长期从事科学与工程计算研究，研究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，现（曾）为SIAM Journal on Numerical Analysis、SIAM Journal of Scientific Computing和IMA Journal of Numerical Analysis等国际知名期刊编委。
