Stucchio, Chris; Soffer, Avy Multiscale resolution of shortwave-longwave interaction. (English) Zbl 1155.65084 Commun. Pure Appl. Math. 62, No. 1, 82-124 (2009). Summary: 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 \(k_{\min}\). Cited in 4 Documents MSC: 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) Keywords:numerical examples; error bounds; high frequencies; shortwaves; low frequencies; longwaves; spectral methods; multiscale algorithm; Schrödinger equation Software:FFTW PDFBibTeX XMLCite \textit{C. Stucchio} and \textit{A. Soffer}, Commun. Pure Appl. Math. 62, No. 1, 82--124 (2009; Zbl 1155.65084) Full Text: DOI arXiv References: [1] Abramowitz, M.; Stegun, I. A., eds. Handbook of mathematical functions with formulas, graphs, and mathematical tables. 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