# Implicit Runge-Kutta processes

@article{Butcher1964ImplicitRP, title={Implicit Runge-Kutta processes}, author={John C. Butcher}, journal={Mathematics of Computation}, year={1964}, volume={18}, pages={50-64} }

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 solution which may be found by iteration (see Appendix). t It will be assumed throughout that f (y) and all its derivatives exist and are continuous so that the Taylor expansions for y and y may be terminated at any term with an error of the same order as the first term omitted. 50

## 619 Citations

### Equilibria of Runge-Kutta methods

- Mathematics
- 1990

SummaryIt is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of…

### Characterization of non-linearly stable implicit Runge-Kutta methods

- Computer Science, Mathematics
- 1982

This paper introduces to the theory of algebraically stable (A-contractive, B-stable) Runge-Kutta methods, methods for which the numerical solutions remain contractive if the (nonlinear) differential equation has contractive solutions.

### On the $BN$ stability of the Runge-Kutta methods

- Computer Science, Mathematics
- 1981

In this note the sufficient conditions that let a RungeKutta s stages method of at least order s be stable are given and a result that has already been demonstrated in another way about the BN stability of implicit Runge-KUTta methods of maximum order has been obtained as a corollary.

### Some relationships between implicit Runge-Kutta, collocation and Lanczosτ methods, and their stability properties

- Mathematics
- 1969

In this paper relationships between various one-step methods for the initial value problem in ordinary differential equations are discussed and a unified treatment of the stability properties of the…

### Contractivity of Runge-Kutta methods

- Mathematics
- 1991

In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value…

### Rational approximations by implicit Runge-Kutta schemes

- Mathematics
- 1970

AbstractEhle [3] has pointed out that then-stage implicit Runge-Kutta (IRK) methods due to Butcher [1] areA-stable in the definition of Dahlquist [2] because they effect the operationR(Ah) whereR(μ)…

### Galerkin/Runge-Kutta discretizations for semilinear parabolic equations

- Computer Science, Mathematics
- 1990

A new class of fully discrete Galerkin/Runge–Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems and offers arbitrarily high-order convergence without suffering from what has been called order reduction.

### A special family of Runge-Kutta methods for solving stiff differential equations

- Computer Science, Mathematics
- 1978

A family of methods of Implicit Runge-Kutta Methods is constructed and some results concerning their maximum attainable order and stability properties are given.

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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…

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- Mathematics
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where y(x) denotes the solution of the differential equation. The idea is to use a quadrature formula to estimate the integral of (1). This requires knowledge of the integrand at specified arguments…

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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.

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