Schwarz waveform relaxation method for one dimensional Schr{\"o}dinger equation with general potential

TL;DR

This paper develops a scalable Schwarz Waveform Relaxation method for solving 1D Schr{"o}dinger equations with various potentials, significantly reducing computation time and improving scalability through novel algorithms and boundary conditions.

Contribution

Introduces a new scalable SWR algorithm for 1D Schr{"o}dinger equations with general potentials, utilizing preconditioning and innovative boundary conditions.

Findings

Algorithm is robust and scalable up to 500 subdomains.

Preconditioning improves efficiency for time-dependent and nonlinear potentials.

Numerical comparisons show reduced computation time.

Abstract

In this paper, we apply the Schwarz Waveform Relaxation (SWR) method to the one dimensional Schr{\"o}dinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schr{\"o}dinger equation with time independent linear potential, which is robust and scalable up to 500 subdo-mains. It reduces significantly computation time compared with the classical algorithms. Concerning the case of time dependent linear potential or the non-linear potential, we use a preprocessed linear operator for the zero potential case as preconditioner which lead to a preconditioned algorithm. This ensures high scalability. Besides, some newly constructed absorbing boundary conditions are used as the transmission condition and compared numerically.