Extended explicit Bel'tyukov pairs of orders 4 and 5 for Volterra integral equations of the second kind

• Published in: Applied numerical mathematics Vol. 34; no. 2; pp. 261 - 274
• Main Authors: Sharp, P.W., Verner, J.H.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 01-07-2000; Elsevier
• Subjects: Bel'tyukov methods; Exact sciences and technology; Integral equations; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis in abstract spaces; Numerical analysis. Scientific computation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Runge–Kutta explicit pairs; Sciences and techniques of general use; Volterra integral equations; Runge–Kutta explicit pairs; Volterra integral equations; Bel'tyukov methods; Butcher table; Bel'tyukov method; Integral equation; Explicit pair; Runge Kutta method; Volterra equation; Embedded method
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/S0168-9274(99)00132-4

The derivation of extended explicit Bel'tyukov pairs of methods for Volterra integral equations of the second kind is related to that of explicit Runge–Kutta pairs, but is more intricate. Techniques previously developed for deriving families of explicit Runge–Kutta pairs and a family of Bel'tyukov pairs of orders 3 and 4 are adapted to construct a parametric family of pairs of orders 4 and 5. While ten stages (of which one is reused) are required in this construction, it is an open question whether pairs of this type and order with fewer stages exist. A range of parameters for which the propagating method satisfies the kernel condition is given, and numerical performance of a representative pair is compared with that of a near-optimal 3,4 pair.