Improved uniform error bound on the time-splitting method for the long-time dynamics of the fractional nonlinear Schrödinger equation

Abstract

We establish the improved uniform error bound on the time-splitting Fourier pseudospectral (TSFP) method for the long-time dynamics of the generalized fractional nonlinear Schrödinger equation (FNLSE) with $O(\varepsilon^2)$-nonlinearity, where $\varepsilon \in (0,1]$ is a dimensionless parameter. Numerically, we discretize the FNLSE by the second-order Strang splitting method in time and Fourier pseudospectral method in space. Combining with energy method, we utilize the regularity compensation oscillation (RCO) technique to rigorously prove the improved uniform error bound at $O(h^{m_0} + \varepsilon^2 \tau^2)$ with the mesh size $h$ and time step $\tau$ up to the long-time at $O(1 / \varepsilon^2)$, which gains an additional $\varepsilon^2$ in time compared with classical error estimates. The key idea behind the RCO technique is to analyze low frequency modes by phase cancellation and control high frequency modes by the regularity of the exact solution. With the help of the RCO technique, we relax some constraints in the previous proof for the improved uniform error bound and extend the result to more general cases. Finally, numerical examples are provided to confirm our improved uniform error bound and demonstrate its suitability in different cases.