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 (long waves, 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 _maxL log(k _max/k _min) log(k _maxL)). Here, L is the width of the region in which the algorithm resolves all frequencies and is independent of k _min.