Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity

TL;DR

This paper analyzes semi-discrete schemes for the nonlinear Schrödinger equation, demonstrating convergence under lower regularity conditions and establishing first and second order accuracy in fractional Sobolev spaces.

Contribution

It introduces a class of time discretization schemes that achieve convergence with less regularity requirements than classical methods, with proven error bounds and numerical validation.

Findings

Convergence under lower regularity assumptions.

First and second order convergence in fractional Sobolev spaces.

Numerical experiments confirm theoretical results.

Abstract

We analyse a class of time discretizations for solving the nonlinear Schr\"odinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \le 3$. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space $H^{r}$, $r \ge 0$, beyond the more typical $L^2$ or $H^\sigma (\sigma>\frac{d}{2}$) -error analysis. Numerical experiments illustrate our results.