Unconditionally stable methods for second order differential equations

Abstract

We use the concept of order stars (see [1]) to prove and generalize a recent result of Dahlquist [2] on unconditionally stable linear multistep methods for second order differential equations. Furthermore a result of Lambert-Watson [3] is generalized to the multistage case. Finally we present unconditionally stable Nyström methods of order 2s (s=1,2,...) and an unconditionally stable modification of Numerov's method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them

Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

“…Following this approach, Fehlberg [6] constructed a set of RKN pairs of orders p( p + 1) (i.e., order p for the step continuation and order p + 1 for the reference method), for p up to 8. Other embedded RKN methods have been derived by Dormand and Prince [5] (order 7(6), requiring 9 stages), Fehlberg, Filippi and Graf [8] (order 1()(11), requiring 17 stages), Filippi and Graf [9] (order 11 (12), requiring 20 stages), and, by the same authors [10], methods of order p + 1( p ), where p runs from 7 to 10 and the corresponding number of stages runs from 9 to 17).…”

“…Correctors having this property are easily obtained from high-order Runge-Kutta methods for first-order ODEs in the following way [12] [3]). They possess a high order, relative to the number of stages, i.e., p = 2s and p = 2s -1, respectively.…”

“…Total number of steps (rejected ones) Table 10 Performance of the parallel code PIRKN, the sequential code DOPRIN, and some additional methods when applied to problem (3.3) Code PIRKN 9(10) pair from [7] 10(11) pair from [8] 11 ( 12) Performance of the parallel code PIRKN, the sequential code DOPRIN, and an additional method when applied to problem (3.4) Code PIRKN 11(12) pair from [ Results for the last test problem are found only in [9]. Together with the PIRKN and DOPRIN results, they are listed in Table 11.…”

Stability of collocation-based Runge-Kutta-Nyström methods

“…For example, the diagonal elements of the Pade table associated with exp(w) satisfy this condition, and hence, the s-stage Gauss-Legendre methods generates-stage, ?-stable RKN methods with stage orders and step point order 2s (cf. [6]). …”

“…!Rd, t 0 :::; t :::; T Our motivation for studying implicit RKN methods is the arrival of parallel computers which enables us to solve the implicit relations occurring in the stage vector equation quite efficiently, so that, what is so far considered as the main disadvantage of fully implicit RKN methods, is reduced a great deal. We consider two types of collocation methods for second-order equations: methods based on direct collocation and on indirect collocation (that is, methods obtained by writing the special second-order equation in first-order form and by applying collocation methods for first-order equations [6]). The theory of indirect collocation methods *) These investigations were supported by the University of Amsterdam who provided the third author with a research grant for spending a total of two years in Amsterdam.…”