Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of L2 compared to classical results in dimension d, which are limited to higher-order (sufficiently smooth) Sobolev spaces Hs with s>d/2. In particular, we are able to establish a global error estimate in L2 for H1 solutions which is roughly of order τ12+5−d12 in dimension d≤3 (τ denoting the time discretization parameter). This breaks the “natural order barrier” of τ1/2 for H1 solutions which holds for classical numerical schemes (even in combination with suitable filter functions).