A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime

We propose and analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for solving the Klein--Gordon (KG) equation with a dimensionless parameter 0<ε≤1 which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e., 0<ε1, the solution of the KG equation propagates waves with amplitude at $O(1)$ and wavelength at O(ε²) in time and $O(1)$ in space, which causes significantly numerical burdens due to the high oscillation in time. The MTI-FP method is designed by adapting a multiscale decomposition by frequency (MDF) to the solution at each time step and applying an exponential wave integrator to the nonlinear Schrödinger equation with wave operator under well-prepared initial data for ε²-frequency and $O(1)$-amplitude waves and a KG-type equation with small initial data for the reminder waves in the MDF. We rigorously establish two independent error bounds in H²-norm to the MTI-FP method at O(hm₀+τ²+ε²) and O(hm₀+τ²/ε²) with h mesh size, τ time step, and m₀≥2 an integer depending on the regularity of the solution, which immediately imply that the MTI-FP 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 confirm the error bounds and demonstrate the efficiency and accuracy of the MTI-FP method for the KG equation, especially in the nonrelativistic limit regime.