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 = 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 Schrödinger equation, but generally applicable) which solves the problem with cost (in space and time) O(kmaxL log(kmax/kmin) log(kmaxL)). Here, L is the width of the region in which the algorithm resolves all frequencies, and is independent of kmin.