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

TL;DR

This work develops a two-level numerical framework based on Localized Orthogonal Decomposition (LOD) for efficiently simulating ground states and dynamics of single-component Bose–Einstein condensates governed by the Gross–Pitaevskii equation. It introduces two fully discrete schemes: a damped inverse-iteration method in the LOD space for ground states and a continuous Galerkin in time (cG-LOD) approach for time evolution with a conservative-like energy in a modified sense, leveraging a projected nonlinearity to curb computational cost. The authors establish key approximation properties of ideal and practical LOD spaces for both the ground-state problem and time invariants, design efficient nonlinear treatment via a sparse 3-valence tensor, and demonstrate high-order spatial accuracy (up to O($H^6$)) and favorable efficiency across 1D–3D numerical experiments, including comparisons to spectral and standard FEM methods. The results indicate substantial potential for scalable, accurate BEC simulations, with notable opportunities for further speedups through newer LOD variants and localized basis constructions.

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.

Figures (7)

• Figure 6.1: Convergence plot of values in Table \ref{['MinEnergyTable']} as measured logarithmically in error versus mesh size $H$
• Figure 6.2: Convergence to ground state of method \ref{['ideal-damped-inverse-iteration']}, initial guess is a normalized Gaussian
• Figure 6.3: Density plot of minimizer, $|u_{\text{gs}}^{_{\text{\tiny LOD}}}|^2$. At $H=1.0$, $l=3$, the difference in accuracy is more than one order of magnitude.
• Figure 6.4: Comparison of different methods in terms of convergence and accuracy per CPU times. The blue line with downward pointing triangles in Figures \ref{['conv_discont']} through \ref{['Total_time_discont']} show the performance of the canonical LOD. The blue line with upwards pointing triangles shows the performance of incorporating the discontinuity in the LOD-space. The choice of $h$ and $\ell$ is according to the triplets $(H,H/h,\ell)$: (1.2,25,1); (1,25,3);(6/7,25,4);(3/4,25,5);(0.6,40,6);(0.5,40,6);(0.4,50,8).
• Figure 6.5: Log-log plot of error versus coarse mesh size $H$ in the case of harmonic potential.

Theorems & Definitions (2)

• Remark 2.1
• Remark 3.1: Well-prepared initial values for optimal accuracy