Stability and dynamical properties of Rosenau-Hyman compactons using Pade approximants

TL;DR

This paper develops a systematic method using Padé approximants for higher-order derivatives to analyze the stability and dynamics of Rosenau-Hyman compactons, including collisions and shock formations, with improved accuracy and reduced errors.

Contribution

It introduces a novel differencing scheme based on Padé approximants for higher derivatives, enhancing stability analysis and simulation accuracy of compactons.

Findings

The most accurate approximation reduces roundoff errors in stability studies.

Different derivative approximation choices affect spurious radiation during compacton scattering.

Fourth-order methods with sixth-order accuracy improve simulation fidelity.

Abstract

We present a systematic approach for calculating higher-order derivatives of smooth functions on a uniform grid using Pad\'e approximants. We illustrate our findings by deriving higher-order approximations using traditional second-order finite-differences formulas as our starting point. We employ these schemes to study the stability and dynamical properties of K(2,2) Rosenau-Hyman (RH) compactons including the collision of two compactons and resultant shock formation. Our approach uses a differencing scheme involving only nearest and next-to-nearest neighbors on a uniform spatial grid. The partial differential equation for the compactons involves first, second and third partial derivatives in the spatial coordinate and we concentrate on four different fourth-order methods which differ in the possibility of increasing the degree of accuracy (or not) of one of the spatial derivatives to sixth order. A method designed to reduce roundoff errors was found to be the most accurate approximation in stability studies of single solitary waves, even though all derivates are accurate only to fourth order. Simulating compacton scattering requires the addition of fourth derivatives related to artificial viscosity. For those problems the different choices lead to different amounts of "spurious" radiation and we compare the virtues of the different choices.