A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition

TL;DR

This paper introduces two fully-discrete numerical methods based on Localized Orthogonal Decomposition for efficiently computing ground states and dynamics of Bose-Einstein condensates, with detailed implementation and performance analysis.

Contribution

The paper develops two novel LOD-based numerical approaches tailored for BECs, optimizing computational efficiency and accuracy for ground states and dynamics simulations.

Findings

Methods demonstrate high convergence rates.

Approaches outperform spectral and finite element methods.

Efficient in 1D, 2D, and 3D simulations.

Abstract

In this work, we consider the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1d, 2d and 3d that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches.