Geodesic
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

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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   ( ε 2 ) 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  h m 0 + τ 2 ε 2 and h m 0 + τ 2 + ε 2 , where  h is the mesh size,  τ is the time step and  m 0 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   ( τ 2 ) in the regimes when either in the regimes when either  ε   =   0   ( 1 ) or 0   <   ε ≲ τ . Numerical results are reported to demonstrate that our error estimates are optimal and sharp.

DOI : 10.1051/m2an/2018015
Classification : 35Q40, 65M70, 65N35, 81W05
Keywords: 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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     author = {Cai, Yongyong and Wang, Yan},
     title = {A uniformly accurate {(UA)} multiscale time integrator pseudospectral method for the nonlinear {Dirac} equation in the nonrelativistic limit regime},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {543--566},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {2},
     year = {2018},
     doi = {10.1051/m2an/2018015},
     zbl = {1404.35377},
     mrnumber = {3834435},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018015/}
}
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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

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