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 < ε 1, the solution of the KG equation propagates waves with amplitude at O(1) and wavelength at O(ε 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 Schrödinger equation with wave operator under well-prepared initial data for ε 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 m 0 +τ 2 +ε 2 ) and O(h m 0 +τ 2 /ε 2 ) with h mesh size, τ 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(τ ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O(τ 2 ) 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.

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter 0 < ε ≤ 1. In fact, the solution to this equation propagates waves with wavelength at O(ε 2 ) when 0 < ε ≪ 1, which brings significantly numerical burdens in practical computation. We rigorously establish two independent error bounds for the two MTIs at O(τ 2 /ε 2 ) and O(ε 2 ) for ε ∈ (0, 1] with τ > 0 as step size, which imply that the two MTIs converge uniformly with linear convergence rate at O(τ) for ε ∈ (0, 1] and optimally with quadratic convergence rate at O(τ 2 ) in the regimes when either ε = O(1) or 0 < ε ≤ τ. Thus the meshing strategy requirement (or ε-scalability) of the two MTIs is τ = O(1) for 0 < ε ≪ 1, which is significantly improved from τ = O(ε 3 ) and τ = O(ε 2 ) requested by finite difference methods and exponential wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards better understanding on the convergence and resolution properties of the two MTIs. In addition, numerical results support the two error bounds very well.

Abstract. An exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0 < ε ≤ 1 and 0 < γ ≤ 1 which are inversely proportional to the plasma frequency and the speed of sound, respectively. The main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space. The method is explicit, symmetric in time, and it is of spectral accuracy in space and second-order accuracy in time for any fixed ε = ε 0 and γ = γ 0 . In the O(1)-plasma frequency and speed of sound regime, i.e., ε = O(1) and γ = O(1), we establish rigorously error estimates for the numerical method in the energy space H 1 × L 2 . We also study numerically the resolution of the method in the simultaneous high-plasma-frequency and subsonic limit regime, i.e., (ε, γ) → 0 under ε γ. In fact, in this singular limit regime, the solution of the KGZ system is highly oscillating in time, i.e., there are propagating waves with wavelength of O(ε 2 ) and O(1) in time and space, respectively. Our extensive numerical results suggest that, in order to compute "correct" solutions in the simultaneous high-plasma-frequency and subsonic limit regime, the meshing strategy (or ε-scalability) is time step τ = O(ε 2 ) and mesh size h = O(1) independent of ε. Finally, we also observe numerically that the method has the property of near conservation of the energy over long time in practical computations.

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