A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with energy-preservation for long-time dynamics of the nonlinear Klein-Gordon equation

Abstract

For the nonlinear Klein-Gordon (NKG) equation with weak nonlinearity characterized by a small parameter ε∈(0,1], the numerical methods for long-time dynamics have received more and more attention. For the NKG equation up to the time at O(1/ε2), the error bounds of classical numerical methods depend significantly on the small parameters ε which creates serious numerical burdens as ε→0+. Recently, an exponential wave integrator Fourier pseudo-spectral (EWIFP) method for the NKG equation has been proposed (Feng et al., 2021) which is uniformly accurate about ε and of course performs well over the classical methods. However, the EWIFP method can not preserve the discrete energy, which is an important structural feature of the true solution in the long-time dynamics from the perspective of geometric numerical integration. In addition, the EWIFP method only is conditionally stable under specific stability condition and the authors did not give a convergence analysis for the method without any CFL condition restrictions on the grid ratio. In this work, we propose an energy-preserving EWIFP (EPEWIFP) method. The proposed method is proved to be time symmetric, unconditionally stable and preserves the discrete energy. We carry out a rigorously error analysis and give uniform error bounds of the method at O(hm0+ε2−βτ2) up to the time at O(1/εβ) with β∈[0,2], mesh size h, time step τ and an integer m0 determined by the regularity conditions. In general, the NKG equation with weak nonlinearity can be converted to an oscillatory NKG equation with wavelength at O(ε2) in time. It is easy to extend the error bounds and energy-preservation properties to the oscillatory NKG equation. Numerical results confirm our error bounds and energy-preservation properties.