Mapped Chebyshev pseudospectral method for the study of multiple scale phenomena

In the framework of mapped pseudospectral methods, we use a polynomial-type mapping function in order to describe accurately the dynamics of systems developing small size structures. Using error criteria related to the spectral interpolation error, the 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 Schrödinger equations and modeling beam propagation in an atomic coherent media, whose spatial sizes differ up to several orders of magnitude. It is demonstrated, also by numerical simulations, that the accuracy properties of the polynomial-type mapping outperform by orders of magnitude the ones of the other studied mapping functions.