Computation of the Interval of Stability of Runge–Kutta–Nyström Methods

Abstract

We present a Mathematica package to compute the interval of stability of Runge-KuttaNyström methods for y = f (t, y).

Stability analysis of frequency and step length dependent Runge–Kutta–Nyström methods

“…Following [1,12], we proceed through computer algebra and analyze the modulus of the eigenvalues of the stability matrix (6), by studying the change of sign of the trace and the determinant of (6). More in details, the interval of stability is the interval of ν 2 values for which [14] | det(M(ν 2 ; θ))| < 1, |trace(M(ν 2 ; θ))| < det(M(ν 2 ; θ)) + 1.…”

“…In Section 3 we describe the symbolicnumerical package which we have developed to analyze the stability of exponential-fitting methods, following the lines already drawn in [1,12]. We believe that computer algebra provides a powerful tool in the analysis of numerical methods, because of its ability to compute expressions, leaving the elements of a computation unevaluated.…”

“…This peculiarity is useful in the construction of the quantities involved in a stability analysis, such as those described in Section 2. Moreover it is possible to perform an analysis on a whole family, when not all the parameters have assumed a constant value, but still some degrees of freedom remain in the choice of parameters, in order to decide which value has to be chosen in order to maximize a certain property (see for instance [11,12]). …”