A Fully Discrete Low-Regularity Integrator for the Nonlinear Schrödinger Equation

Abstract

For the solution of the one dimensional cubic nonlinear Schrödinger equation on the torus, we propose and analyze a fully discrete low-regularity integrator. The considered scheme is explicit. Its implementation relies on the fast Fourier transform with a complexity of $${\mathcal {O}}(N\log N)$$ operations per time step, where N denotes the degrees of freedom in the spatial discretization. We prove that the new scheme provides an $${\mathcal {O}}(\tau ^{\frac{3}{2}\gamma -\frac{1}{2}-\varepsilon }+N^{-\gamma })$$ error bound in $$L^2$$ for any initial data in $$H^\gamma $$ , $$\frac{1}{2}<\gamma \le 1$$ , where $$\tau $$ denotes the temporal step size. Numerical examples illustrate this convergence behavior.