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Multiscale resolution of shortwave‐longwave interaction
Author(s) -
Stucchio Chris,
Soffer Avy
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20258
Subject(s) - wavenumber , longwave , shortwave , mathematics , impurity , computational physics , domain (mathematical analysis) , resolution (logic) , space (punctuation) , binary logarithm , physics , mathematical analysis , algorithm , radiative transfer , optics , quantum mechanics , computer science , artificial intelligence , operating system
In the study of time‐dependent waves, it is computationally expensive to solve a problem in which high frequencies (shortwaves, with wavenumber k = k max ) and low frequencies (longwaves, near k = k min ) mix. Consider a problem in which low frequencies scatter off a sharp impurity. The impurity generates high frequencies that propagate and spread throughout the computational domain, while the domain must be large enough to contain several longwaves. Conventional spectral methods have a computational cost that is proportional to O ( k max / k min log( k max / k min )). We present here a multiscale algorithm (implemented for the Schrödinger equation but generally applicable) that solves the problem with cost (in space and time) O ( k max L log( k max / k min ) log( k max L )). Here, L is the width of the region in which the algorithm resolves all frequencies and is independent of k min . © 2008 Wiley Periodicals, Inc.