Mapped Chebyshev pseudospectral method to study multiple scale phenomena

TL;DR

This paper introduces a new polynomial mapping function for mapped Chebyshev pseudospectral methods, significantly improving accuracy in modeling systems with almost singular structures and multi-scale phenomena.

Contribution

A novel polynomial-type mapping function is developed and validated, outperforming previous mappings in accuracy for studying collapse, shock waves, and multi-scale coupled beam dynamics.

Findings

The new mapping outperforms previous methods by orders of magnitude in accuracy.

Numerical simulations confirm improved modeling of collapse and shock phenomena.

Enhanced precision in simulating coupled nonlinear Schrödinger equations.

Abstract

In the framework of mapped pseudospectral methods, we introduce a new polynomial-type mapping function in order to describe accurately the dynamics of systems developing almost singular structures. Using error criteria related to the spectral interpolation error, the new polynomial-type mapping is compared against previously proposed mappings for the study of collapse and shock wave phenomena. As a physical application, we study the dynamics of two coupled beams, described by coupled nonlinear Schr\"odinger equations and modeling beam propagation in an atomic coherent media, whose spatial sizes differs up to several orders of magnitude. It is demonstrated, also by numerical simulations, that the accuracy properties of the new polynomial-type mapping outperforms in orders of magnitude the ones of the other studied mapping functions.