Domain decomposition algorithms for the two dimensional nonlinear Schr{\"o}dinger equation and simulation of Bose-Einstein condensates

TL;DR

This paper develops and tests advanced domain decomposition algorithms, including a preconditioned Schwarz method, for efficiently solving the 2D nonlinear Schrödinger equation and simulating Bose-Einstein condensates, demonstrating improved convergence and reduced computation time.

Contribution

The paper introduces a preconditioned Schwarz algorithm and new transmission conditions for better efficiency in simulating Bose-Einstein condensates.

Findings

Preconditioned algorithm improves convergence rate.

Reduces computation time in simulations.

Absorbing transmission condition enhances performance.

Abstract

In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schr{\"o}dinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.