Self-similar Radiation from Numerical Rosenau-Hyman Compactons

TL;DR

This paper investigates the self-similar radiation emitted by numerical simulations of Rosenau-Hyman compactons, revealing its numerical origin and analyzing its properties across different numerical methods and parameters.

Contribution

It introduces a self-similar scaling ansatz for the radiation from compactons and demonstrates its consistency across multiple numerical schemes.

Findings

Radiation amplitude decreases with grid refinement.

Radiation exhibits exponential decay over time.

Self-similar scaling describes the radiation profile.

Abstract

The numerical simulation of compactons, solitary waves with compact support, is characterized by the presence of spurious phenomena, as numerically-induced radiation, which is illustrated here using four numerical methods applied to the Rosenau-Hyman K(p,p) equation. Both forward and backward radiations are emitted from the compacton presenting a self-similar shape which has been illustrated graphically by the proper scaling. A grid refinement study shows that the amplitude of the radiations decreases as the grid size does, confirming its numerical origin. The front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters. The amplitude of the radiations decreases exponentially in time, being characterized by a nearly constant scaling exponent. An ansatz for both the backward and forward radiations corresponding to a self-similar function characterized by the scaling exponent is suggested by the present numerical results.