# 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} }

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

## 386 Citations

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### Problem #11: generation of Runge-Kutta equations

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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.

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### Generation and application of the equations of condition for high order Runge-Kutta methods

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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…

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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…

### On Runge-Kutta processes of high order

- MathematicsJournal of the Australian Mathematical Society
- 1964

An (explicit) Runge-Kutta process is a means of numerically solving the differential equation , at the point x = x0+h, where y, f may be vectors.

### B-Series Analysis of Stochastic Runge-Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

- Computer Science, MathematicsSIAM J. Numer. Anal.
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A unifying approach is employed for the construction of stochastic B-series which is valid both for It o- and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method.

### 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

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- 1987

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.

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