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

摘要

We present a systematic approach for calculating higher-order derivatives ofsmooth functions on a uniform grid using Pad\'e approximants. We illustrate ourfindings by deriving higher-order approximations using traditional second-orderfinite-differences formulas as our starting point. We employ these schemes tostudy the stability and dynamical properties of K(2,2) Rosenau-Hyman (RH)compactons including the collision of two compactons and resultant shockformation. Our approach uses a differencing scheme involving only nearest andnext-to-nearest neighbors on a uniform spatial grid. The partial differentialequation for the compactons involves first, second and third partialderivatives in the spatial coordinate and we concentrate on four differentfourth-order methods which differ in the possibility of increasing the degreeof accuracy (or not) of one of the spatial derivatives to sixth order. A methoddesigned to reduce roundoff errors was found to be the most accurateapproximation in stability studies of single solitary waves, even though allderivates are accurate only to fourth order. Simulating compacton scatteringrequires 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.