Volume 16, Issue 2
On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

Xuanchun Dong, Zhiguo Xu & Xiaofei Zhao

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

Published online: 2014-08

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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 ε∈(0,1]. In the nonrelativistic limit regime, the small ε produces high oscillations in exact solutions with wavelength of O(ε2) 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 ε = 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(ε2)-oscillatory solutions when 0 < ε ≪ 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic O(ε−2∆t2) temporal discretization error bound (∆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(ε−4∆t2). 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.

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@Article{CiCP-16-440, author = {}, title = {On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime}, journal = {Communications in Computational Physics}, year = {2014}, volume = {16}, number = {2}, pages = {440--466}, 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 ε∈(0,1]. In the nonrelativistic limit regime, the small ε produces high oscillations in exact solutions with wavelength of O(ε2) 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 ε = 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(ε2)-oscillatory solutions when 0 < ε ≪ 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic O(ε−2∆t2) temporal discretization error bound (∆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(ε−4∆t2). 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.280813.190214a}, url = {http://global-sci.org/intro/article_detail/cicp/7049.html} }
TY - JOUR T1 - On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime JO - Communications in Computational Physics VL - 2 SP - 440 EP - 466 PY - 2014 DA - 2014/08 SN - 16 DO - http://doi.org/10.4208/cicp.280813.190214a UR - https://global-sci.org/intro/article_detail/cicp/7049.html KW - AB -

In this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameter ε∈(0,1]. In the nonrelativistic limit regime, the small ε produces high oscillations in exact solutions with wavelength of O(ε2) 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 ε = 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(ε2)-oscillatory solutions when 0 < ε ≪ 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic O(ε−2∆t2) temporal discretization error bound (∆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(ε−4∆t2). 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.

Xuanchun Dong, Zhiguo Xu & Xiaofei Zhao. (2020). On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime. Communications in Computational Physics. 16 (2). 440-466. doi:10.4208/cicp.280813.190214a
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