Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems

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

“…, 1) T . It is assumed that M (z) has complex conjugate eigenvalues for sufficiently small values of z (Van der Houwen and Sommeijer, 1989). This assumption produces an oscillatory numerical solution, whose behaviour depends on the eigenvalues of M (z), which is called the stability matrix.…”

“…RKN methods for which results P (z) ≡ 1 are zero-dissipative methods: the dissipation error for them is null (Van der Houwen and Sommeijer, 1989).…”

“…We study the linear stability properties of a RKN method for ODEs (1.1) (Sharp et al, 1990; Van der Houwen and Sommeijer, 1989;Van der Houwen et al, 1991). From Van der Houwen et al (1991), we recall briefly some definitions that will be useful in the following.…”

“…Furthermore simplification of expressions involving unevaluated elements can be performed. These features can be used in the derivation of new numerical methods for ODEs when not only the order of consistency but many other characteristics, such as the stability properties, the magnitude of the error coefficients or dispersive order (Van der Houwen and Sommeijer, 1989) must be taken into account at the same time.…”

“…Then, we can use the unevaluated coefficients to heighten other properties that we consider valuable. If we are interested in methods for secondorder ODEs with oscillatory solutions, we should consider, for example, the dispersive order ( Van der Houwen and Sommeijer, 1989). After solving the dispersive relations (Van der Houwen and Sommeijer, 1989) in a symbolic way too, we can determine the last free parameters applying numerical optimization to minimize the error coefficients, and computing the stability interval of the resulting methods.…”