A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Dirac Equation in the Nonrelativistic Limit Regime

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the (linear) Dirac equation with a dimensionless parameter ε ∈ (0, 1] which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e., 0 < ε 1, the solution exhibits highly oscillatory propagating waves with wavelength O(ε2) and O(1) in time and space, respectively. Due to the rapid temporal oscillation, designing and analyzing numerical methods with uniform error bounds in ε ∈ (0, 1] is quite challenging. 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 hm0 + τ 2 ε2 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 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(τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O(τ2) in the regimes when either ε = O(1) or 0 < ε τ . 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 ε → 0+.