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Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime


Authors: Weizhu Bao and Chunmei Su
Journal: Math. Comp. 87 (2018), 2133-2158
MSC (2010): Primary 35Q55, 65M06, 65M12, 65M15
DOI: https://doi.org/10.1090/mcom/3278
Published electronically: November 22, 2017
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Abstract: We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov (KGZ) system with a dimensionless parameter $ \varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., $ 0<\varepsilon \ll 1$, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with $ O(\varepsilon )$-wavelength in time and $ O(1)$-wavelength in space as well as outgoing initial layers in space at speed $ O(1/\varepsilon )$. This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ system. By applying an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at $ O(h^2+\tau ^2/\varepsilon )$ and $ O(h^2+\tau +\varepsilon )$ with $ h$ mesh size and $ \tau $ time step. Thus we obtain a uniform error bound at $ O(h^2+\tau )$ for $ 0<\varepsilon \le 1$. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and $ \varepsilon $-dependent error bounds between the solutions of KGZ system and its limiting model when $ \varepsilon \to 0^+$. Finally, numerical results are reported to confirm our error bounds.


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Additional Information

Weizhu Bao
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
Email: matbaowz@nus.edu.sg

Chunmei Su
Affiliation: Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
Address at time of publication: Department of Mathematics, National University of Singapore, Singapore 119076
Email: sucm@csrc.ac.cn

DOI: https://doi.org/10.1090/mcom/3278
Keywords: Klein-Gordon-Zakharov system, subsonic limit, highly oscillatory, uniform error bound, finite difference method, asymptotic consistent formulation
Received by editor(s): December 30, 2016
Received by editor(s) in revised form: March 18, 2017
Published electronically: November 22, 2017
Additional Notes: The first author was supported by Singapore Ministry of Education Academic Research Fund Tier 2 R-146-000-223-112.
The second author is the corresponding author. The second author was supported by Natural Science Foundation of China Grant U1530401 and the Postdoctoral Science Foundation of China Grant 2016M600904.
Article copyright: © Copyright 2017 American Mathematical Society

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