Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equation

The aim of this paper is to establish improved uniform error bounds under \(\varvec{H^{\alpha /2}}\) ${\mathit{H}}^{\mathit{\alpha }\mathbf{/}\mathbf{2}}$ -norm \(\varvec{(1<\alpha \le 2)}\) $\mathbf{(}\mathbf{1}\mathbf{<}\mathit{\alpha }\mathbf{\le }\mathbf{2}\mathbf{)}$ for the long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation (NSFSGE) by a Lawson-type exponential integrator Fourier pseudo-spectral (LEI-FP) method. Firstly, a Lawson-type exponential integrator method is used to discretize the time direction. Then, the Fourier pseudo-spectral method is applied to discretize the space direction. We rigorously prove that the equation is energy conservation in a continuous state. Regularity compensation oscillation (RCO) technique is employed to strictly prove the improved uniform error bounds at \(\varvec{O\left( \varepsilon ^2 \tau \right) }\) $\mathit{O}\left({\mathit{\epsilon }}^{\mathbf{2}},\mathit{\tau }\right)$ in temporal semi-discretization and \(\varvec{O\left( h^m+\varepsilon ^2 \tau \right) }\) $\mathit{O}\left({\mathit{h}}^{\mathit{m}},\mathbf{+},{\mathit{\epsilon }}^{\mathbf{2}},\mathit{\tau }\right)$ in full-discretization up to the long-time \(\varvec{T_{\varepsilon }=T / \varepsilon ^2}\) ${\mathit{T}}_{\mathit{\epsilon }}\mathbf{=}\mathit{T}\mathbf{/}{\mathit{\epsilon }}^{\mathbf{2}}$ ( \(\varvec{T>0}\) $\mathit{T}\mathbf{>}\mathbf{0}$ fixed), respectively. To obtain the convergence order \(\varvec{h^{m}}\) ${\mathit{h}}^{\mathit{m}}$ in space, we only need to directly prove it instead of proving that the numerical solution is \(\varvec{H^{m+\alpha /2}}\) ${\mathit{H}}^{\mathit{m}\mathbf{+}\mathit{\alpha }\mathbf{/}\mathbf{2}}$ -norm bounded as before. Complex NSFSGE and oscillatory NSFSGE are also discussed. This is the novel work to construct the improved uniform error bounds for the long-time dynamics of the high-dimensional nonlinear space fractional Klein-Gordon equation with non-polynomial nonlinearity. Finally, numerical examples in two-dimension and three-dimension are provided to confirm the improved error bounds, and we find drastically different evolving patterns between NSFSGE and the classical sine-Gordon equation.