Improved error bounds of an energy-preserving exponential wave integrator for the long-time weakly nonlinear Klein–Gordon equation

Abstract We investigate numerical methods for the long-time dynamics of the weakly nonlinear Klein–Gordon equation (NKGE) with a quadratic nonlinearity ($\varepsilon ^{p}u^{p+1}$, $p=1$), where the nonlinear strength is characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. Different from previous studies, we consider a Gautschi-type exponential wave integrator Fourier pseudo-spectral method for the NKGE, which is time symmetric and energy-preserving. For the quadratic power nonlinearity ($p=1$), we establish improved error bounds for the proposed numerical scheme at $O(h^{m} + \varepsilon \tau ^{2})$ up to the long time $O(1/\varepsilon )$, where $m$ depends on the regularity of the exact solution. Here and below, $h$ is the spatial mesh size and $\tau$ is the time step size. This is in contrast to the classic cubic nonlinearity case $p=2$, where only uniform error bounds $O(h^{m} + \tau ^{2})$ hold up to the long-time $O(1/\varepsilon ^{p})$. The regularity compensation oscillation technique is highly involved in the error analysis, which has been developed recently to analyze the accumulation of errors carefully. For the first time, we report that the improved error bounds hold for Gautschi-type methods, while only time splitting methods and Deuflhard-type (or Lawson-type) exponential wave integrators are known to enjoy improved error bounds (w.r.t. $\varepsilon$) in literature. Our results indicate that the improved error bounds are not only related to numerical discretizations, but also related to the nonlinear structure. Numerical experiments are presented to verify theoretical findings.