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On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

Published online by Cambridge University Press:  03 June 2015

Xuanchun Dong,
Zhiguo Xu and
Xiaofei Zhao
Xuanchun Dong*
Affiliation:
Beijing Computational Science Research Center, Beijing 100084, P.R. China
Zhiguo Xu*
Affiliation:
Beijing Computational Science Research Center, Beijing 100084, P.R. ChinaCollege of Mathematics, Jilin University, Changchun 130012, P.R. China
Xiaofei Zhao*
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore
*
Email:dong.xuanchun@gmail.com
Email:xuzg11@csrc.ac.cn
Corresponding author.Email:zhxfnus@gmail.com
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Abstract

In this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameter ɛ ∊ (0,1]. In the nonrelativistic limit regime, the small ɛ produces high oscillations in exact solutions with wavelength of (ɛ2) in time. The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time, with both the nonlinear and linear subproblems exactly integrable in time and, respectively, Fourier frequency spaces. The method is fully explicit and time reversible. Moreover, we establish rigorously the optimal error bounds of a second-order TSFP for fixed ɛ = (1), thanks to an observation that the scheme coincides with a type of trigonometric integrator. As the second task, numerical studies are carried out, with special efforts made to applying the TSFP in the nonrelativistic limit regime, which are geared towards understanding its temporal resolution capacity and meshing strategy for (ɛ2)-oscillatory solutions when 0 < ɛ « 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic (ɛ−2Δt2) temporal discretization error bound (Δt refers to time step). On the other hand, the temporal error bounds for most trigonometric integrators, such as the well-established Gautschi-type integrator in, are (ɛ−4Δt2). Thus, our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous or numerical, are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.

Keywords

35L7065M1265M1565M70Klein-Gordon equationhigh oscillationtime-splittingtrigonometric integratorerror estimatemeshing strategy.

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Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014, pp. 440 - 466
Copyright
Copyright © Global Science Press Limited 2014

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