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 Hα /2 -norm (1<α ≤ 2) 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 O( ε ² τ ) in temporal semi-discretization and O( hᵐ+ε ² τ ) in full-discretization up to the long-time Tε=T / ε ² ( T>0 fixed), respectively. To obtain the convergence order hᵐ in space, we only need to directly prove it instead of proving that the numerical solution is Hm+α /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.