A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime

Cai, Yongyong; Wang, Yan

doi:10.1051/m2an/2018015

Cai, Yongyong ¹ ; Wang, Yan ¹

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 543-566.

Résumé

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 $ε \in (0, 1]$ 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 $ε \in (0, 1]$ . 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 $h^{m_{0}} + \frac{τ^{2}}{ε^{2}}$ 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 $ε \in (0, 1]$ 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.

MR Zbl

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

Affiliations des auteurs :

Cai, Yongyong ¹ ; Wang, Yan 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://www.numdam.org/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. http://www.numdam.org/articles/10.1051/m2an/2018015/

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