New optimized Schwarz algorithms for one dimensional Schr\"odinger equation with general potential

TL;DR

This paper introduces new optimized Schwarz algorithms for the 1D Schrödinger equation with general potentials, offering direct and preconditioned methods that are scalable, robust, and more efficient than classical iterative approaches.

Contribution

The paper develops a direct Schwarz algorithm for time-independent linear potentials and a preconditioned algorithm for time-dependent or nonlinear potentials, improving efficiency and scalability.

Findings

The direct algorithm is free of transmission condition choices.

Preconditioned algorithm converges independently of transmission conditions.

Algorithms are scalable up to 256 subdomains and reduce computation time.

Abstract

The aim of this paper is to develop new optimized Schwarz algorithms for the one dimensional Schr{\"o}dinger equation with linear or nonlinear potential. After presenting the classical algorithm which is an iterative process, we propose a new algorithm for the Schr{\"o}dinger equation with time-independent linear potential. Thanks to two main ingredients (constructing explicitly the interface problem and using a direct method on the interface problem), the new algorithm turns to be a direct process. Thus, it is free to choose the transmission condition. Concerning the case of time-dependent linear potential or nonlinear potential, we propose to use a pre-processed linear operator as preconditioner which leads to a preconditioned algorithm. Numerically , the convergence is also independent of the transmission condition. In addition, both of these new algorithms implemented in parallel cluster are robust, scalable up to 256 sub domains (MPI process) and take much less computation time than the classical one, especially for the nonlinear case.