Uniform error bounds of time-splitting spectral methods for the long-time dynamics of the nonlinear Klein–Gordon equation with weak nonlinearity
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- Math. Comp. 91 (2022), 811-842 Request permission
Abstract:
We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein–Gordon equation (NKGE) with weak power-type nonlinearity and $O(1)$ initial data, while the nonlinearity strength is characterized by $\varepsilon ^{p}$ with a constant $p \in \mathbb {N}^+$ and a dimensionless parameter $\varepsilon \in (0, 1]$, for the long-time dynamics up to the time at $O(\varepsilon ^{-\beta })$ with $0 \leq \beta \leq p$. In fact, when $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of NKGE with small initial data and $O(1)$ nonlinearity strength, while the amplitude of the initial data (and the solution) is at $O(\varepsilon )$. By reformulating the NKGE into a relativistic nonlinear Schrödinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at $O(h^{m}+\varepsilon ^{p-\beta }\tau ^2)$ of the TSFP method with $h$ mesh size, $\tau$ time step and $m\ge 2$ depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at $O(\varepsilon ^{-\beta })$ and uniformly valid for $\varepsilon \in (0,1]$. Especially, the error bounds are uniformly at the second-order rate for the large time step $\tau = O(\varepsilon ^{-(p-\beta )/2})$ in the parameter regime $0\le \beta <p$. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon ^{\beta })$ in time and wave velocity at $O(\varepsilon ^{-\beta })$.References
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Additional Information
- Weizhu Bao
- Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
- MR Author ID: 354327
- ORCID: 0000-0003-3418-9625
- Email: matbaowz@nus.edu.sg
- Yue Feng
- Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
- ORCID: 0000-0002-3895-3283
- Email: fengyue@u.nus.edu
- Chunmei Su
- Affiliation: Yau Mathematical Sciences Center, Tsinghua University, 100084 Beijing, People’s Republic of China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, People’s Republic of China
- Email: sucm@tsinghua.edu.cn
- Received by editor(s): October 9, 2020
- Received by editor(s) in revised form: May 14, 2021, and August 11, 2021
- Published electronically: October 28, 2021
- Additional Notes: This work was partially supported by the Ministry of Education of Singapore grant R-146-000-290-114 (the first and second authors) and Alexander von Humboldt Foundation (the third author)
The second author is the corresponding author. - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 811-842
- MSC (2020): Primary 35L70, 65M12, 65M15, 65M70, 81-08
- DOI: https://doi.org/10.1090/mcom/3694
- MathSciNet review: 4379977