A second-order low-regularity integrator for the nonlinear Schrödinger equation

AbstractIn this paper, we analyze a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the d dimensional torus $\mathbb{T}^{d}$

T
d

. The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76, 2022). It is explicit and efficient to implement. Here, we present an alternative derivation and give a rigorous error analysis. In particular, we prove the second-order convergence in $H^{\gamma }(\mathbb{T}^{d})$

H
γ

(

T
d

)
for initial data in $H^{\gamma +2}(\mathbb{T}^{d})$

H

γ
+
2


(

T
d

)
for any $\gamma > d/2$
γ
>
d
/
2
. This improves the previous work (Knöller et al. in SIAM J. Numer. Anal. 57:1967–1986, 2019).The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.