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Science China Mathematics

, Volume 59, Issue 8, pp 1461–1494 | Cite as

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

  • WeiZhu BaoEmail author
  • YongYong Cai
  • XiaoWei Jia
  • Jia Yin
Articles

Abstract

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(ε 3) and \(h = O\left( {\sqrt \varepsilon } \right)\). 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(ε 2) and h = O(1) when 0 < ε ≤ 1. Extensive numerical results are reported to confirm our error estimates.

Keywords

nonlinear Dirac equation nonrelativistic limit regime Crank-Nicolson finite difference method exponential wave integrator time splitting spectral method ε-scalability 

MSC(2010)

35Q55 65M70 65N12 65N15 81Q05 

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • WeiZhu Bao
    • 1
    Email author
  • YongYong Cai
    • 2
    • 3
  • XiaoWei Jia
    • 1
  • Jia Yin
    • 4
  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Beijing Computational Science Research CenterBeijingChina
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA
  4. 4.NUS Graduate School for Integrative Sciences and Engineering (NGS)National University of SingaporeSingaporeSingapore

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