Multiscale resolution of shortwave‐longwave interaction

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.