Completely Imbedded Runge-Kutta Pairs

• Published in: SIAM journal on numerical analysis Vol. 31; no. 4; pp. 1169 - 1190
• Main Authors: Sharp, P. W., Verner, J. H.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01-08-1994
• Subjects: Applied mathematics; Approximation; Butchering; Coefficients; Engineering research; Error rates; Exact sciences and technology; Iterated integrals; Mathematics; Methods; New family; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Polynomials; Research grants; Runge Kutta method; Sciences and techniques of general use; Embedding; Error estimation; Runge Kutta method; Differential equation; Lagrange interpolation; Initial value problem
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0731061

Recently, pairs of explicit Runge-Kutta methods of orders 5 and 6 based on a new design have been derived independently by several authors. These pairs may be implemented so that the approximation of order 6 may be propagated using eight stages, and the extra derivative evaluation required for the error-estimating approximation of order 5 may be used again in the next step. An improved derivation of these pairs leads to a convenient generalization and thus to the derivation of families of higher-order pairs of the same design. For arbitrary p, pairs of orders p - 1 and p require s ≤ (p2 - 7p + 22)/2 internal stages (the inequality holds for p ≥ 8) and the derivative evaluation of the propagating stage. Furthermore, for each p, imposing an additional constraint on the nodes removes the necessity for the extra derivative evaluation. For each pair of this restricted subfamily, the method of order p utilizes all s stages so that the method of order p - 1 is properly imbedded. Examples of both types of new pairs are displayed, and these are compared numerically to some known pairs.