Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

Abstract

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.

An Exponential Wave Integrator Sine Pseudospectral Method for the Klein--Gordon--Zakharov System

“…Based on our results, we will show that the ε-scalability of the EWI-SP method for the KGZ system in the simultaneous high-plasma-frequency and subsonic limit regime is τ = O(ε 2 ) and h = O (1). In addition, we also observe that the EWI-SP method nearly conserves the total energy over a long time in practical computation for the dynamics of the KGZ system, which is a favorable property for numerical time integrators and has been extensively studied for second-order ordinary different equations (ODEs) and dispersive partial differential equations (PDEs) in the literatures; see, e.g., [4,5,10,11,12] and references therein.…”

“…Based on our results [5], for the KG equation in the nonrelativistic limit regime, i.e., 0 < ε 1, in order to compute "correct" solutions, the frequently used finite difference time domain (FDTD) methods [1,14,24,30] share the same ε-scalability: time step τ = O(ε 3 ) and mesh size h = O(1) [5]. In addition, our results demonstrated that the exponential wave integrator sine pseudospectral (EWI-SP) method for the KG equation can improve the ε-scalability to τ = O(1) and τ = O(ε 2 ) for linear and nonlinear KG equation, respectively [5]. This suggests that, when the solution has a highly oscillatory nature in time, the EWI-SP method has much better temporal resolution than the FDTD methods.…”

“…Here we present an exponential wave integrator sine spectral/pseudospectral discretization for (2.1)-(2.5), which applies the sine spectral/pseudospectral discretization to spatial derivatives followed by using an exponential wave integrator (EWI) [4,5,18,19,21,22] for temporal discretization in phase space.…”

“…In addition, the solution (ψ, φ) of the KGZ system propagates waves with wavelength O(ε 2 ) and O(ε), respectively, in time when 0 < ε 1 under ε γ. This highly oscillatory nature in time brings severe numerical burdens, which is similar to the case of the numerical solution for the Klein-Gordon (KG) equation in the nonrelativistic limit regime [5], making the computation in the simultaneous high-plasma-frequency and subsonic limit regime extremely challenging. To our knowledge, so far there are few results on the numerics of the KGZ system in this regime.…”

“…The main aim of this paper is to extend the EWI-SP method for discretizing the KG equation [5] to the KGZ system; establish rigorously error estimates for the method in the energy space H 1 × L 2 when ε = O(1) and γ = O(1); and study numerically the resolution capacity and ε-scalability of the method in the simultaneous high-plasma-frequency and subsonic limit regime. Based on our results, we will show that the ε-scalability of the EWI-SP method for the KGZ system in the simultaneous high-plasma-frequency and subsonic limit regime is τ = O(ε 2 ) and h = O (1).…”