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 (ε2) 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 inε ∈ (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 hm0+τ2/ε2andhm0 + τ2 + ε2, where his the mesh size, τis the time step and m0depends 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 (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.
Uniform error bounds of exponential wave integrator methods for the long-time dynamics of the Dirac equation with small potentials
