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

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