A fully discrete low-regularity integrator for the nonlinear Schrödinger equation

08/10/2021

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by   Alexander Ostermann, et al.

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For the solution of the cubic nonlinear Schrödinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform with a complexity of 𝒪(Nlog N) operations per time step, where N denotes the degrees of freedom in the spatial discretisation. We prove that the new scheme provides an 𝒪(τ^3/2γ-1/2-ε+N^-γ) error bound in L^2 for any initial data belonging to H^γ, 1/2<γ≤ 1, where τ denotes the temporal step size. Numerical examples illustrate this convergence behavior.