Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

作     者：BAO WeiZhu CAI YongYong JIA XiaoWei YIN Jia 

作者机构：Department of MathematicsNational University of SingaporeSingapore 119076Singapore Beijing Computational Science Research CenterBeijing 100193China Department of MathematicsPurdue UniversityWest LafayetteIN 47907USA NUS Graduate School for Integrative Sciences and Engineering(NGS)National University of SingaporeSingapore 117456Singapore 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2016年第59卷第8期

页      面：1461-1494页

核心收录：

学科分类：07[理学] 070102[理学-计算数学] 0701[理学-数学] 

基　　金：supported by the Ministry of Education of Singapore(Grant No.R146-000-196-112) National Natural Science Foundation of China(Grant No.91430103) 

主　　题：nonlinear Dirac equation nonrelativistic limit regime Crank-Nicolson finite difference method exponential wave integrator time splitting spectral method ^-scalability 

摘      要：We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0 〈 ε〈〈1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O（ ε^2） and O（1） in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference （CNFD） method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ- as well as the small parameter 0 〈 ε≤1 Based on the error bound, in order to obtain ＇correct＇ numerical solutions in the nonrelativistic limit regime, i.e., 0 〈 ε≤1 , the CNFD method requests the ε-scalability： τ- = O（ε3） and h = O（√ε）. Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time- splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O（ε2） and h = O（1） when 0 〈 ε 〈〈 1. Extensive numerical results are reported to confirm our error estimates.