Commun. Comput. Phys., 16 (2014), pp. 440-466.

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

Xuanchun Dong 1, Zhiguo Xu 2, Xiaofei Zhao 3*

1 Beijing Computational Science Research Center, Beijing 100084, P.R. China.
2 College of Mathematics, Jilin University, Changchun 130012, P.R. China; Beijing Computational Science Research Center, Beijing 100084, P.R. China.
3 Department of Mathematics, National University of Singapore, Singapore 119076, Singapore.

Received 28 August 2013; Accepted (in revised version) 19 February 2014
Available online 23 May 2014


In this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameter $\varepsilon\in(0,1]$. In the nonrelativistic limit regime, the small $\varepsilon$ produces high oscillations in exact solutions with wavelength of $O(\varepsilon^2)$ 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 $\varepsilon O(1)$, 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 $O(\varepsilon^2)$-oscillatory solutions when $0<\varepsilon \ll 1$. It suggests that the method has uniform spectral accuracy in space, and an asymptotic $O(\varepsilon^{-2}{\Delta t^2})$ temporal discretization error bound ($\Delta t$ 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 [6], are $O(\varepsilon^{-4}{\Delta t^2})$. 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.

AMS subject classifications: 35L70, 65M12, 65M15, 65M70

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Key words: Klein-Gordon equation, high oscillation, time-splitting, trigonometric integrator, error estimate, meshing strategy.

*Corresponding author.
Email: (X. Dong), (X. Zhao), (Z. Xu)

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