Improved uniform error bounds of exponential wave integrator method for long-time dynamics of the space fractional Klein-Gordon equation with weak nonlinearity

TL;DR

This paper develops an improved uniform error bound for a numerical method solving the long-time dynamics of the nonlinear space fractional Klein-Gordon equation, demonstrating enhanced accuracy and stability over extended periods.

Contribution

It introduces an improved error analysis for a second-order exponential wave integrator combined with Fourier spectral methods, applicable to complex NSFKGE with oscillatory nonlinearities.

Findings

Established an $O(h^m+ au^2)$ error bound in $H^{rac{eta}{2}}$-norm.

Validated the theoretical results through numerical experiments.

Extended analysis to complex and oscillatory NSFKGE cases.

Abstract

An improved uniform error bound at $O\left(h^m+\varepsilon^2 \tau^2\right)$ is established in $H^{\alpha/2}$-norm for the long-time dynamics of the nonlinear space fractional Klein-Gordon equation (NSFKGE). A second-order exponential wave integrator (EWI) method is used to semi-discretize NSFKGE in time and the Fourier spectral method in space is applied to derive the full-discretization scheme. Regularity compensation oscillation (RCO) technique is employed to prove the improved uniform error bounds at $O\left(\varepsilon^2 \tau^2\right)$ in temporal semi-discretization and $O\left(h^m+\varepsilon^2 \tau^2\right)$ in full-discretization up to the long-time $T_{\varepsilon}=T / \varepsilon^2$ ($T>0$ fixed), respectively. Complex NSFKGE and oscillatory complex NSFKGE with nonlinear terms of general power exponents are also discussed. Finally, the correctness of the theoretical analysis and the effectiveness of the method are verified by numerical examples.