Phase Lag Sensitivity Analysis for Numerical Integration

TL;DR

This paper introduces a new technique for exponential fitting in numerical integration that improves accuracy for oscillatory functions by ensuring the phase lag's first derivatives vanish at the fitted frequency.

Contribution

It proposes a novel method based on the vanishing of phase lag derivatives, enhancing the performance of exponential fitting techniques in oscillatory problems.

Findings

Methods show improved phase lag behavior.

Enhanced accuracy in oscillatory numerical integration.

Theoretical proof of improved characteristics.

Abstract

In the field of numerical integration, methods specially tuned on oscillating functions, are of great practical importance. Such methods are needed in various branches of natural sciences, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. Among others, probably the most important tool used to construct efficient methods for oscillatory problems is the exponential (trigonometric) fitting. The basic characteristic of these methods is that their phase lag vanishes at a predefined frequency. In this work, we introduce a new tool which improves the behavior of exponentially fitted numerical methods. The new technique is based on the vanishing of the first derivatives of the phase lag function at the fitted frequency. It is proved in the text that these methods present improved characteristics in oscillatory problems.