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

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 $ (u,{\mathrm{\partial }}_tu)\in {L}^{\mathrm{\infty }}(0,T;{H}^{1+\frac{d}{4}}\times {H}^{\frac{d}{4}})$ . In one dimension, the proposed method is shown to have almost $ \frac{4}{3}$ -order convergence in L ∞ (0, T; H 1 × L 2 ) 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.