A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 543-566.

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<ε1 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+τ2ε2 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 in the regimes when either ε=0(1) or 0<ετ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.

DOI : 10.1051/m2an/2018015
Classification : 35Q40, 65M70, 65N35, 81W05
Mots-clés : Nonlinear Dirac equation, nonrelativistic limit, uniformly accurate, multiscale time integrator, exponential wave integrator, spectral method, error bound
Cai, Yongyong 1 ; Wang, Yan 1

1
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     title = {A uniformly accurate {(UA)} multiscale time integrator pseudospectral method for the nonlinear {Dirac} equation in the nonrelativistic limit regime},
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     publisher = {EDP-Sciences},
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     zbl = {1404.35377},
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Cai, Yongyong; Wang, Yan. A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 543-566. doi : 10.1051/m2an/2018015. http://www.numdam.org/articles/10.1051/m2an/2018015/

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