Methods for numerical simulation of NLS

Abstract: The advent of supercomputers made it possible to model multidimensional NLS and revealed new problems: new parallelizable algorithms were required.

For equations of the "parabolic" type, which include the non-stationary Schrödinger equation, numerical schemes have very stringent stability conditions: $\Delta t < \Delta x^2$, which, in fact, slows down the solution of the problem when the grid is refined. In addition, in equations of the NLSE type, high spatial harmonics do not decay with time, but have rapidly changing phases, which leads even under a "relatively mild" condition of stability to the phenomenon of random phases.

A review of grid and spectral methods for finding approximate solutions of the NSE is given, and the possibilities of using the FFT are analyzed. The problem of increasing the counting step with respect to time and typical errors are discussed. Brief reviews of the use of the operator exponential method and the method of nonreflecting boundary conditions are given. The possibilities of the hyperbolization method for NLS are discussed.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods