A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation

¹ 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

^* Corresponding author: buyang.li@polyu.edu.hk; libuyang@gmail.com

Received: 2 May 2022
Accepted: 22 November 2022

Abstract

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.