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

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, ∂t u) ∈ L∞ (0, T; H1+d/4 × Hd/4). In one dimension, the proposed method is shown to have almost 4/3-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.