Some Extended Explicit Bel'Tyukov Pairs for Volterra Integral Equations of the Second Kind

• Published in: SIAM journal on numerical analysis Vol. 38; no. 2; pp. 347 - 359
• Main Authors: Sharp, P.W., Verner, J.H.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2001
• Subjects: Applied mathematics; Approximation; Coefficients; Councils; Differential equations; Engineering research; Exact sciences and technology; Information technology; Integral equations; Integral equations, integral transforms; Mathematical vectors; Mathematics; Matrices; New family; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Orthogonality; Research universities; Runge Kutta method; Sciences and techniques of general use; Tableaux; Minimum; Butcher tableau; Tolerance; Integral; Integral equation; Beltyukov pair; Explicit pair; Runge Kutta method; Volterra equation; Volterra integral equations; Runge Kutta pair
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S003614299631280X

We derive and investigate a family of pairs of extended explicit Bel'tyukov Runge-Kutta (EBVRK) formulas to treat Volterra integral equations of the second kind. Each pair uses six stages and consists of an order 3 formula completely embedded in an order 4 formula. We examine the structure of EBVRK pairs to facilitate their derivation, and in particular, we show that at least six stages are needed to form such pairs. This design is motivated to allow a variable stepsize in order to satisfy a prescribed error tolerance with a minimum of computation. Some numerical examples illustrate the procedures.