Superconvergence of time invariants for the Gross-Pitaevskii equation

TL;DR

This paper introduces a high-accuracy, structure-preserving numerical scheme for the Gross-Pitaevskii equation, combining Crank-Nicolson time discretization with localized orthogonal decomposition for spatial discretization, achieving superior invariant conservation.

Contribution

It develops a novel numerical method that preserves time invariants of the Gross-Pitaevskii equation with high accuracy, integrating Crank-Nicolson and localized orthogonal decomposition techniques.

Findings

Achieves $ ext{O}(H^6)$ accuracy in invariant conservation.

Proves $L^{ abla}( ext{infinity})(L^2)$-norm approximation order $ ext{O}( au^2 + H^4)$.

Demonstrates computational efficiency through numerical experiments.

Abstract

This paper considers the numerical treatment of the time-dependent Gross-Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture the time invariants with an accuracy of order $\mathcal{O}(H^6)$ with respect to the chosen mesh size $H$. This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the $L^{\infty}(L^2)$-norm with order $\mathcal{O}(\tau^2 + H^4)$, where $\tau$ denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.