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

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.