Optimized integrating factor technique for Schr\"odinger-like equations

TL;DR

This paper introduces an optimized integrating factor technique for Schrödinger-like equations that leverages gauge freedom to enhance computational efficiency, especially beneficial for long-term simulations.

Contribution

It develops an adaptive gauge optimization method for the integrating factor technique, significantly improving speed in solving Schrödinger equations.

Findings

Speed increases by a factor of five or more.

Effective for nonlinear Schrödinger and Schrödinger-Newton equations.

Enhances long-term simulation efficiency.

Abstract

The integrating factor technique is widely used to solve numerically (in particular) the Schr\"odinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge condition of the potential. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We illustrate this approach with the nonlinear Schr\"odinger (NLS) and with the Schr\"odinger-Newton (SN) equations. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. This gain is crucial for long time simulations.