A frequency-dependent $p$-adaptive technique for spectral methods

TL;DR

This paper introduces a frequency-dependent p-adaptive spectral method that dynamically adjusts the expansion order based on oscillation frequency, improving the numerical solution of problems like the Schrödinger equation in unbounded domains.

Contribution

The paper presents a novel frequency-dependent p-adaptive technique for spectral methods, enhancing their ability to handle oscillations and unbounded domain problems.

Findings

Successfully captures oscillatory behavior in Schrödinger equation

Adapts expansion order based on frequency indicator

Handles diffusion, advection, and oscillations effectively

Abstract

When using spectral methods, a question arises as how to determine the expansion order, especially for time-dependent problems in which emerging oscillations may require adjusting the expansion order. In this paper, we propose a frequency-dependent $p$-adaptive technique that adaptively adjusts the expansion order based on a frequency indicator. Using this $p$-adaptive technique, combined with recently proposed scaling and moving techniques, we are able to devise an adaptive spectral method in unbounded domains that can capture and handle diffusion, advection, and oscillations. As an application, we use this adaptive spectral method to numerically solve the Schr\"{o}dinger equation in the whole domain and successfully capture the solution's oscillatory behavior at infinity.