Superconvergence of time invariants for the Gross–Pitaevskii equation

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 and allows to capture the time invariants with an accuracy of order O ( H 6 ) \mathcal {O}(H^6) with respect to the chosen mesh size H 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 ∞ ( L 2 ) L^{\infty }(L^2) -norm with order O ( τ 2 + H 4 ) \mathcal {O}(\tau ^2 + H^4) , where τ \tau denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

On optimal convergence rates for discrete minimizers of the Gross-Pitaevskii energy in LOD spaces

“…The following approximation properties of the LOD space are well known and are explicitly stated and proved in [31,Section 2.1].…”

“…Since then, it has been further developed to suit a range of different problems, e.g., parabolic equations [36,35,37], equations describing wave phenomena, see, e.g., [1,41,28,21,34], non-linear problems [26,44], and more. In the context of nonlinear Schrödinger equations and GPEs, LOD techniques have been suggested in [27,31]. For a review of the LOD we refer to the textbook by Målqvist and Peterseim [39] and the recent survey article on numerical homogenization [3].…”

“…In such a setting, the dynamics are described by nonlinear Schrödinger equations (NLS), where the initial values are typically ground states of the GPE (with respect to some modified configuration). In [31] it was recently proved that when the NLS is discretized with LOD spaces, then the energy is approximated and conserved with an accuracy of order O(H 6 ), provided that this accuracy can be already guaranteed for the initial value. Hence, this is exactly what we are establishing in this paper, as it justifies that the ground states computed in LOD-spaces can be straightforwardly used as initial values in an NLS without sacrificing the accuracy with which the energy is approximated over time.…”

Operator Compression with Deep Neural Networks

“…The LOD was introduced in [MP14] and theoretically and practically works for very general coefficients. It has also been successfully applied to other problem classes, for instance wave propagation problems in the context of Helmholtz and Maxwell equations [Pet17, GP15, GHV18, RHB19, MV20] or the wave equation [AH17, PS17, MP19, GM21], eigenvalue problems [MP15, MP17], and in connection with time-dependent nonlinear Schrödinger equations [HW20]. However, it requires a slight deviation from locality.…”

Uniform $L^\infty$-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

“…The reason is that this would also require higher order regularity of the solutions to the auxiliary problem and also corresponding stability bounds in higher order Sobolev norms. Such stability estimates for the auxiliary problem are however only available in H 2 (D), which would hence only yield optimale convergence rates for P 1 -Lagrange finite elements or for particular generalized FE spaces that exploit only low regularity [36]. In this work we solve this issue by only applying the error splitting technique to obtain L ∞ (L ∞ )-bounds for the discrete approximations, as this does not require higher regularity of the semi-discrete auxiliary solution.…”