# 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

## 616 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.

## References

SHOWING 1-10 OF 23 REFERENCES

Coefficients for the study of Runge-Kutta integration processes

- MathematicsJournal of the Australian Mathematical Society
- 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…

A method for the numerical integration of ordinary differential equations

- Mathematics
- 1958

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…

A process for the step-by-step integration of differential equations in an automatic digital computing machine

- MathematicsMathematical 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.

"J."

- PhilosophyThe New Yale Book of Quotations
- 2021

however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)…

Über die numerische Integration von Differentialgleichungen

- Acta Soc. Sei. Fennicae, v
- 1925

Numerical Analysis, 1st edition, McGraw-Hill

- New York,
- 1957