Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein--Gordon equation with weak nonlinearity

TL;DR

This paper derives improved uniform error bounds for time-splitting numerical methods applied to the long-time simulation of the nonlinear Klein-Gordon equation with weak nonlinearity, enhancing accuracy over extended periods.

Contribution

It introduces a novel analysis using the RCO technique to establish sharper error bounds for splitting methods on NKGE with weak nonlinearity, valid over long times.

Findings

Error bounds of O(ε²τ²) for semi-discretization

Error bounds of O(h^m + ε²τ²) for full discretization

Numerical results confirm the sharpness of the bounds

Abstract

We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0 < \varepsilon \leq 1$ a dimensionless parameter. Actually, when $0 < \varepsilon \ll 1$, the NKGE with $O(\varepsilon^2)$ nonlinearity and $O(1)$ initial data is equivalent to that with $O(1)$ nonlinearity and small initial data of which the amplitude is at $O(\varepsilon)$. We begin with a semi-discretization of the NKGE by the second-order time-splitting method, and followed by a full-discretization via the Fourier spectral method in space. Employing the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the exact solution and analyzes the low frequency modes by phase cancellation and energy method, we carry out the improved uniform error bounds at $O(\varepsilon^2\tau^2)$ and $O(h^m+\varepsilon^2\tau^2)$ for the second-order semi-discretization and full-discretization up to the long time $T_\varepsilon = T/\varepsilon^2$ with $T$ fixed, respectively. Extensions to higher order time-splitting methods and the case of an oscillatory complex NKGE are also discussed. Finally, numerical results are provided to confirm the improved error bounds and to demonstrate that they are sharp.