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- James H. Verner
- 1993

For a particular family of pairs of explicit Runge–Kutta methods of orders $p - 1$ and p, sets of efficient, continuously differentiable interpolants of several orders up to p are characters zed… (More)

- James H. Verner
- 1996

Abstract To illustrate his idea for propagating an approximate solution of an initial value problem, Runge (1895) included a pair of formulas or orders 1 and 2 (a 1,2 pair) whose difference could be… (More)

- James H. Verner
- 1991

In [“The Numerical Analysis of Ordinary Differential Equations,” John Wiley, New York, 1987, pp. 298–303], Butcher derives a family of nine-stage formula pairs of orders 5 and 6. If the fifth-order… (More)

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… (More)

Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the… (More)

- James H. Verner
- Numerical Algorithms
- 2009

Explicit Runge–Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two criteria for selection are proposed with a… (More)

- James H. Verner
- 2013

Dierent types of error-estimating pairs of explicit Runge{Kutta methods can be distinguished by the way the algebraic order-conditions are satised. Here, two new types of pairs are derived. The… (More)

A continuous explicit Runge-Kutta (CERK) method provides a continuous approximation to an initial value problem. Such a method may be obtained by appending additional stages to a discrete method, or… (More)

- James H. Verner
- 2006

In [Japan JIAM 19 (2002) 227], Jackiewicz and Verner derived formulas for, and tested the implementation of two-step Runge-Kutta (TSRK) pairs. For pairs of orders 3 and 4, the error estimator… (More)

- Philip W. Sharp, James H. Verner
- SIAM J. Numerical Analysis
- 2000

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… (More)