In this talk I will introduce a novel time discretization for the Gross-Pitaevskii equation at low-regularity on an arbitrary domain \(\Omega \subset \mathbb{R}^d\), \(d \le 3\), with non-smooth potential. I will first show the construction of our first and second order low-regularity integrators. I will discuss the stability issues which arise during the construction of the second-order low-regularity integrator, and will then propose two different approaches to guarantee the stability of our proposed scheme.

I will also present a local and global \(L^2(\Omega)\) and \(H^1(\Omega)\) error analysis for the first and second low-regularity integrators, and for a class of solutions and potential in appropriate Sobolev spaces.

These new schemes, together with their optimal local error, allow for convergence under lower regularity assumptions than required by classical methods.