A uniformly accurate exponential wave integrator method for the nonlinear Klein–Gordon equation with highly oscillatory potential

The nonlinear Klein–Gordon equation with a highly oscillatory potential (NKGE-OP) frequently occurs in recent studies of some multiscale dynamical systems, where the temporal oscillations causes the major numerical and analycal difficulties. In this paper, we propose a uniformly accurate second-order exponential wave integrator (EWI) method for arbitrary nonlinearity by integrating the potential function exactly twice. The proposed method has a fully explicit and concise form, and thus it can be efficiently implemented by using the fast Fourier transform. We give rigourously error analysis and establish second-order uniform error bounds for the numerical solutions without any CFL-type condition constraint. Moreover, the method is proved to be time symmetric which preserves the time symmetry of the considered system. Numerical experiments prove the correctness of our theoretical analysis and the effectiveness of our method.