# Coefficients for the study of Runge-Kutta integration processes

```@article{Butcher1963CoefficientsFT,
title={Coefficients for the study of Runge-Kutta integration processes},
author={John C. Butcher},
journal={Journal of the Australian Mathematical Society},
year={1963},
volume={3},
pages={185 - 201}
}```
• J. Butcher
• Published 1 May 1963
• Mathematics
• Journal of the Australian Mathematical Society
We consider a set of η first order simultaneous differential equations in the dependent variables y1, y2, …, yn and the independent variable x ⋮ No loss of gernerality results from taking the functions f1, f2, …, fn to be independent of x, for if this were not so an additional dependent variable yn+1, anc be introduced which always equals x and thus satisfies the differential equation
383 Citations
Problem #11: generation of Runge-Kutta equations
The Runge-Kutta method for integrating an autonomous system of ordinary differential equations, k and m as large as possible, and the number of conditions and variables is given.
Generation and application of the equations of condition for high order Runge-Kutta methods
This thesis develops the equations of condition necessary for determining the coefficients for Runge-Kutta methods used in the solution of ordinary differential equations. The equations of condition
Another approach to Runge-Kutta methods
The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the
Pseudo Runge-Kutta
• Mathematics, Computer Science
• 2005
This paper considers only (0. 1) because the numerical formulas for such a system of equations or an equivalent high order single equation are almost similar to those of the scalar equation (00 1).
A new theoretical approach to Runge-Kutta methods
Runge–Kutta (RK)-methods are treated here as linear multistage methods within the author’s concept of A-methods, and it is shown how the order of a RK-method depends on the error constants of its stages.
Order conditions for numerical methods for partitioned ordinary differential equations
SummaryMotivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of “P-series” is studied. This theory yields the general structure of the order conditions for
ON THE INTEGRATION PROCESSES OF A. HUf
• Mathematics
• 2008
to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it. In the derivation of the
Runge-Kutta Type Integration Formulas Including the Evaluation of the Second Derivative Part I
Runge-Kutta methods (RK methods, in short) are popular because of the high accuracy and the feasibility of changing step-size, and in general the methods are expressed as follows.

## References

SHOWING 1-9 OF 9 REFERENCES
A process for the step-by-step integration of differential equations in an automatic digital computing machine
• S. Gill
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 1951
It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known, and one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers.
Aufgaben und Lehrsätze aus der Analysis
• Mathematics
The Mathematical Gazette
• 1926
Aufgaben und LehrstUze aus der Analysis I
• Grand. Math. Wiss
• 1925
t)ber die numerische Integration von Differentialgleichungen
• Ada Soc. Sci. Fennicae 50,
• 1925
Uber die numerische AufUJsung von Differentialgleichungen
• Math . Ann .
• 1895
Aufgaben und LehrstUze aus der Analysis I, p. 301
• Grand. Math. Wiss. 19 (Springer,
• 1925
An operational method for the study of integration processes
• Proceedings of conference on data processing and Automatic Computing Machines at Weapons Research Establishment,
• 1957