Volume 52, Number 2, March–April 2018
|Page(s)||543 - 566|
|Published online||11 July 2018|
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime★
Beijing Computational Science Research Center,
100193, P.R. China
* Corresponding author: e-mail: email@example.com
Accepted: 22 February 2018
A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε2) and O (1) in time and space, respectively. In the nonrelativistic regime, i.e., 0 < ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds in ε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as hm0+τ2/ε2 and hm0 + τ2 + ε2, where h is the mesh size, τ is the time step and m0 depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ2) in the regimes when either ε = O (1) or 0 < ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.
Mathematics Subject Classification: 35Q40 / 65M70 / 65N35 / 81W05
Key words: Nonlinear Dirac equation / nonrelativistic limit / uniformly accurate / multiscale time integrator / exponential wave integrator / spectral method / error bound
© EDP Sciences, SMAI 2018
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