Volume 57, Number 2, March-April 2023
|Page(s)||899 - 919|
|Published online||30 March 2023|
A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
2 Laboratoire Jacques-Louis Lions, Sorbonne Université, Bureau 16-26-315, 4 place Jussieu, Paris, France
Accepted: 22 November 2022
The numerical approximation of nonsmooth solutions of the semilinear Klein–Gordon equation in the d-dimensional space, with d = 1, 2, 3, is studied based on the discovery of a new cancellation structure in the equation. This cancellation structure allows us to construct a low-regularity correction of the Lie splitting method (i.e., exponential Euler method), which can significantly improve the accuracy of the numerical solutions under low-regularity conditions compared with other second-order methods. In particular, the proposed time-stepping method can have second-order convergence in the energy space under the regularity condition . In one dimension, the proposed method is shown to have almost -order convergence in L∞(0, T; H1 × L2) for solutions in the same space, i.e., no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. The numerical experiments show that the proposed time-stepping method is much more accurate than previously proposed methods for approximating the time dynamics of nonsmooth solutions of the semilinear Klein–Gordon equation.
Mathematics Subject Classification: 65M12 / 65M15 / 76D05
Key words: Semilinear Klein–Gordon equation / wave equation / energy space / low regularity / second order / error estimates
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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