Speaker: Weizhu Bao, Professor, Department of Mathematics, National University of Singapore

Inviter: Weiying Zheng, Professor

Title: Multiscale methods and analysis for (nonlinear) Dirac equation

Time & Venue: 2024.04.22 16:00-17:00 N219

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.