We analyze rigourously error estimates and compare numerically temporal/ spatial resolution of various numerical methods for solving the Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter 0 < ε 1 which is inversely proportional to the speed of light. In this regime, the solution 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. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter ε. Based on the error bounds, in order to compute 'correct' solutions when 0 < ε 1, the four FDTD methods share the same ε-scalability: τ = O(ε 3 ). Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their ε-scalability is improved to τ = O(1) and τ = O(ε 2 ) for linear and nonlinear KG equations, respectively, when 0 < ε 1. Numerical results are reported to support our error estimates.

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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