Self-similar radiation from numerical Rosenau–Hyman compactons

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 RosenauHyman 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.

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

“…This scheme is an extension of the scheme introduced first by Rus and Villatoro [33,34]. This gives a fourth-order accurate approximation of the first-order derivative,…”

“…Our derivation recovers as special cases the Padé approximants first introduced by Rus and Villatoro [22,[33][34][35]. We illustrate our approach for the particular case when second-order finite-differences formulas are used as the starting point to derive at least fourthorder accurate approximations of the first three derivatives of a smooth function.…”

“…In Sec. II, we show that the approximation schemes discussed by RV [33,34] can be identified as special cases of a systematic improvement scheme that uses Padé approximants to derive at least fourth-order accurate approximations for the first three spatial derivatives u (we have introduced the spatial discretization x m = mh, u(x) → u m and the roman numeral superscript denotes the order of spatial derivative at x m ) by starting with second-order finite-differences approximations. In general, one can begin with finite-differences approximations of any arbitrary even order, and improve upon these by at least two orders of accuracy by using suitable Padé approximants.…”

Properties of compacton-anticompacton collisions

“…Our derivation contained as special cases the Padé approximants first introduced by Rus and Villatoro [40][41][42].…”

“…m , and u (iii) m , with a sixthorder accurate Padé approximant for the third one, denoted as follows: (6,4,4) -this approximation scheme is an extension of the scheme introduced by Sanz-Serna et al [33,34] using a fourth-order Petrov-Galerkin finiteelement method, and corresponds to a sixth-order accurate third-order derivative, u (iii) m ; (4,6,4) -this scheme features a sixth-order accurate approximation for u (ii) m ; and (4,4,6) -this scheme was introduced first by Rus and Villatoro [40,41] and corresponds to a sixth-order accurate approximation for u (iii) m . We also consider a (4,4,4) scheme that was shown to minimize the extent of the radiation train in our previous K(2, 2) and CSS studies.…”