Multiscale Resolution of Shortwave-Longwave Interaction

TL;DR

This paper introduces a multiscale algorithm for efficiently solving wave problems involving both high and low frequencies, significantly reducing computational costs compared to traditional spectral methods.

Contribution

The paper presents a novel multiscale algorithm applicable to wave equations that reduces computational complexity by decoupling frequency scales, demonstrated on the Schrödinger equation.

Findings

Reduces computational cost from O(kmax/kmin log(kmax/kmin)) to O(kmax L log(kmax / kmin) log(kmax L))

Efficiently handles problems with wide frequency ranges and large domains

Applicable to various wave equations beyond Schrödinger

Abstract

In the study of time-dependent waves, it is computationally expensive to solve a problem in which high frequencies (shortwaves, with wavenumber k = kmax) and low frequencies (longwaves, near k=kmin) mix. Consider a problem in which low frequencies scatter off a sharp impurity. The impurity generates high frequencies which propagate and spread throughout the computational domain, while the domain must be large enough to contain several longwaves. Conventional spectral methods have computational cost proportional to O(kmax/kmin \log (kmax/kmin)). We present here a multiscale algorithm (implemented for the Schrodinger equation, but generally applicable) which solves the problem with cost (in space and time) O(kmax L log(kmax / kmin) \log(kmax L)). Here, L is the width of the region in which the algorithm resolves all frequencies, and is independent of kmin.