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

Abstract

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\epsilon \in (0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\epsilon \ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O({\epsilon }^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\epsilon \in (0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{{\tau }^2}{{\epsilon }^2}$ and $h^{m_0}+{\tau }^2+{\epsilon }^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply 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 $O(\tau )$ for all $\epsilon \in (0,1]$ and optimally with quadratic convergence rate at $O({\tau }^2)$ in the regimes when either $\epsilon =O(1)$ or $0<\epsilon \lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\epsilon \to 0^{+}$.

25 pages, 1 figure

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Cited by in corpus (5)

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• Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces
• A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation
• Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic regime without magnetic potential
• A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials