High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation

TL;DR

This paper introduces a new family of twelve-step multistep methods for solving the Schrödinger equation, improving phase lag properties to enhance accuracy and reduce sensitivity to frequency estimation errors.

Contribution

A novel methodology for constructing multistep methods that minimizes phase lag and its derivatives, leading to more accurate and robust numerical integration of the Schrödinger equation.

Findings

Improved phase lag characteristics demonstrated through error analysis.

Numerical applications confirm enhanced accuracy and robustness.

Reduced sensitivity to frequency estimation errors.

Abstract

In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications.