An Efficient Unconditionally Energy-Stable Numerical Scheme for Bose--Einstein Condensate

TL;DR

This paper introduces a novel explicit numerical scheme for Bose--Einstein condensates that guarantees unconditional energy stability, $L^2$ conservation, and convergence, validated through extensive numerical experiments.

Contribution

The authors develop the first explicit, unconditionally energy-stable numerical method for Bose--Einstein condensates with proven convergence and optimal error estimates.

Findings

The scheme ensures unconditional free energy dissipation.

It conserves the $L^2$ norm at each step.

Numerical results agree well with reference solutions.

Abstract

A numerical framework is proposed and analyzed for computing the ground state of Bose--Einstein condensates. A gradient flow approach is developed, incorporating both a Lagrange multiplier to enforce the $L^2$ conservation and a free energy dissipation. An explicit approximation is applied to the chemical potential, combined with an exponential time differencing (ETD) operator to the diffusion part, as well a stabilizing operator, to obtain an intermediate numerical profile. Afterward, an $L^2$ normalization is applied at the next numerical stage. A theoretical analysis reveals a free energy dissipation under a maximum norm bound assumption for the numerical solution, and such a maximum norm bound could be recovered by a careful convergence analysis and error estimate. In the authors' knowledge, the proposed method is the first numerical work that preserves the following combined theoretical properties: (1) an explicit computation at each time step, (2) unconditional free energy dissipation, (3) $L^2$ norm conservation at each time step, (4) a theoretical justification of convergence analysis and optimal rate error estimate. Comprehensive numerical experiments validate these theoretical results, demonstrating excellent agreement with established reference solutions.