Removing trailing tails and delays induced by artificial dissipation in Pad\'e numerical schemes for stable compacton collisions

TL;DR

This paper presents a method to eliminate artificial dissipation-induced tails and delays in Padé schemes simulating compacton collisions, maintaining stability and accuracy in numerical solutions.

Contribution

It introduces a tail removal technique based on adiabatic perturbation methods to counteract artificial dissipation effects in Padé numerical schemes.

Findings

Complete tail removal achieved with minimal ripple

Numerical stability preserved during compacton collisions

Method effective for various n values in K(n,n) equations

Abstract

The numerical simulation of colliding solitary waves with compact support arising from the Rosenau-Hyman K(n,n) equation requires the addition of artificial dissipation for stability in the majority of methods. The price to pay is the appearance of trailing tails, amplitude damping, and delays as the solution evolves. These undesirable effects can be corrected by properly counterbalancing two sources of artificial dissipation; this procedure is designed by using the slow time evolution of the parameters of the solitary waves under the presence of the dissipation determined by means of adiabatic perturbation methods. The validity of the tail removal methodology is demonstrated on a Pad\'e numerical scheme. The tails are completely removed leaving only a small compact ripple at the original position of their front, and the numerical stability of the scheme under compacton collisions is preserved, as shown by extensive numerical experiments for several values of n.