# A stability property of implicit Runge-Kutta methods

@article{Butcher1975ASP, title={A stability property of implicit Runge-Kutta methods}, author={John C. Butcher}, journal={BIT Numerical Mathematics}, year={1975}, volume={15}, pages={358-361} }

A class of implicit Runge-Kutta methods is shown to possess a stability property which is a natural extension of the notion ofA-stability for non-linear systems.

## 147 Citations

High order algebraically stable Runge-Kutta methods

- Computer Science
- 1978

An analysis is made of the family ofs-stage Runge-Kutta methods of order 2s—2 or more which possesses this property.

On theA-stability of implicit Runge-Kutta processes

- Mathematics
- 1978

A new technique to calculate the characteristic functions and to examine theA-stability of implicit Runge-Kutta processes is presented. This technique is based on a direct algebraic approach and an…

A note onB-stability of Runge-Kutta methods

- Computer Science
- 1980

It is proved that for any irreducible Runge-Kutta method these three stability concepts are equivalent.

Nonlinear stability of multistep multiderivative methods

- Mathematics
- 1990

In this paper we examine nonlinear stability of multistep multiderivative methods for initial value problems of a class K(, in a Banach space. Stability criteria are established which extend results…

Stability of Runge-Kutta methods for stiff ordinary differential equations

- Mathematics, Computer Science
- 1994

This work analyzes the integration of initial value problems for stiff systems of ordinary differential equations by Runge–Kutta methods. The author uses the characterization of stiff initial value…

Non-linear stability of a general class of differential equation methods

- Mathematics, Computer Science
- 1980

For a class of methods sufficiently general as to include linear multistep and Runge-Kutta methods as special cases, a concept known as algebraic stability is defined, based on a non-linear test problem, which enables estimates of error growth to be provided.

The equivalence ofB-stability andA-stability

- Mathematics
- 1984

It is shown to what extent the linear (A-st stability) and non-linear (G-stability) stability concepts are equivalent for multistep methods in their one-leg formulation for Runge-Kutta methods.

On Runge-Kutta methods for parabolic problems with time-dependent coefficients

- Mathematics
- 1986

Analyse des approximations completement discretes de Galerkin pour les equations paraboliques a coefficients dependant du temps. Utilisation des methodes de Runge-Kutta implicites

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The concept of strongA-stability is defined. A class of stronglyA-stable Runge-Kutta processes is introduced. It is also noted that several classes of implicit Runge-Kutta processes defined by Ehle…

Implicit Runge-Kutta processes

- Mathematics
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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…