A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
Résumé
We propose and rigourously analyze a multiscale time integrator Fourier
pseudospectral (MTI-FP) method for the Dirac equation
with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light.
In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits
highly oscillatory propagating waves
with wavelength $O(\varepsilon^2)$ 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 $\varepsilon\in(0,1]$.
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 $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$,
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 $\varepsilon\in(0,1]$ and optimally
with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$
or $0<\varepsilon\lesssim \tau$. 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 $\varepsilon\to0^+$.
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