Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine--Gordon equation

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \varepsilon \ll 1$ a dimensionless parameter up to the time at $O(1/\varepsilon^2)$. The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discreization at $O(\varepsilon^2\tau)$ instead of $O(\tau)$ according to classical error estimates and at $O(h^m+\varepsilon^2\tau)$ for the full-discretization up to the time $T_{\varepsilon} = T/\varepsilon^2$ with $T>0$ fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the NKGE with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with $O(\varepsilon^2)$ wavelength in time and $O(\varepsilon^{-2})$ wave speed, which indicates that the temporal error is independent of $\varepsilon$ when the time step size is chosen as $O(\varepsilon^2)$. Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.