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 < epsilon <= 1 which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e., 0 < epsilon << 1, the solution of the KG equation propagates waves with amplitude at O(1) and wavelength at O(epsilon(2)) 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 Schrodinger equation with wave operator under well-prepared initial data for epsilon(2)-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 H-2-norm to the MTI-FP method at O(h(m0)+tau(2)+epsilon(2)) and O(h(m0)+tau(2)/epsilon(2)) with h mesh size, tau time step, and m(0) >= 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(t) for all epsilon epsilon (0, 1] and optimally with quadratic convergence rate at O(tau(2)) in the regimes when either epsilon = O(1) or 0 < epsilon <= tau . 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.