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The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
- J. Butcher
- Mathematics, Computer Science
- 10 February 1987
TLDR
Implicit Runge-Kutta processes
- J. Butcher
- Mathematics
- 1964
Received November 1, 1962. Revised April 22, 1963. * If the function f(y) satisfies a Lipschitz condition and h is sufficiently small, then the equations defining g(1>, g(2), • • • , gw have a unique…
Stability Criteria for Implicit Runge–Kutta Methods
- K. Burrage, J. Butcher
- Mathematics, Economics
- 1 February 1979
A comparison is made of two stability criteria. The first is a modification to nonautonomous problems of A-stability and the second is a similar modification of B-stability. It is shown that under…
Numerical methods for ordinary differential equations
- J. Butcher
- Computer Science
- 2003
TLDR
Coefficients for the study of Runge-Kutta integration processes
- J. Butcher
- MathematicsJournal of the Australian Mathematical Society
- 1 May 1963
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…
An algebraic theory of integration methods
- J. Butcher
- Mathematics, Computer Science
- 1972
A class of integration methods which includes Runge-Kutta methods, as well as the Picard successive approximation method, is shown to be related to a certain group which can be represented as the…
On Runge-Kutta processes of high order
- J. Butcher
- MathematicsJournal of the Australian Mathematical Society
- 1 May 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.
Non-linear stability of a general class of differential equation methods
- K. Burrage, J. Butcher
- Mathematics, Computer Science
- 1 June 1980
TLDR
General linear methods for ordinary differential equations
- J. Butcher
- Computer ScienceMath. Comput. Simul.
- 20 July 2009
On the Convergence of Numerical Solutions to Ordinary Differential Equations
- J. Butcher
- Mathematics
- 1966
Numerical methods for the solution of the initial value problem ill ordinary differential equations fall mainly into two categories: multi-step methods and RungeKutta methods. For these and for some…
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