Journal ArticleDOI
Non-linear stability of a general class of differential equation methods
Kevin Burrage,John C. Butcher +1 more
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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.Abstract:
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. This property is based on a non-linear test problem and extends existing results on Runge-Kutta methods and on linear multistep and one-leg methods. The algebraic stability properties of a number of particular methods in these families are studied and a generalization is made which enables estimates of error growth to be provided for certain classes of methods.read more
Citations
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Difference methods for differential inclusions: a survey
Asen L. Dontchev,Frank Lempio +1 more
TL;DR: The main objective of this survey is to study convergence properties of difference methods applied to differential inclusions to present a number of results scattered in the literature.
Journal ArticleDOI
EXPINT---A MATLAB package for exponential integrators
TL;DR: A MATLAB1 package is described which aims to facilitate the quick deployment and testing of exponential integrators, of Runge--Kutta, multistep, and general linear type, along with several well-known examples.
Journal ArticleDOI
Diagonally-implicit multi-stage integration methods
TL;DR: The aim of the presented paper is to select from the large family of possible general linear methods, just a single class which has cosiderable potential for efficient implementation.
Journal ArticleDOI
Non‐linear B‐stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity
Juan C. Simo,Sanjay Govindjee +1 more
TL;DR: In this article, a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward-Euler return maps as a particular case.
References
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Journal ArticleDOI
A special stability problem for linear multistep methods
TL;DR: The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property as discussed by the authors, and error bounds are derived which are valid under rather general conditions.
Journal ArticleDOI
Stability Criteria for Implicit Runge–Kutta Methods
Kevin Burrage,John C. Butcher +1 more
TL;DR: In this paper, a comparison is made of two stability criteria, i.e., A-stability and Bstability, and it is shown that under certain mild conditions these two concepts are equivalent.
Journal ArticleDOI
On the Butcher group and general multi-value methods
Ernst Hairer,Gerhard Wanner +1 more
TL;DR: This paper proves a theorem (“Theorem 6”) on the composition of the Butcher series, shown to be fundamental for the theory of Runge-Kutta methods, and extends the multi-value methods of J. Butcher to the multiderivative case, which leads to a big class of integration methods for ordinary differential equations, including the methods of Nordsieck and Gear.
Journal ArticleDOI
G-stability is equivalent toA-stability
TL;DR: In this article, it was shown that G-stability is equivalent to A-stableness, and that a Liapunov function exists if the stability region of the method contains a circle (halfplane), provided that the system satisfies a monotonicity condition related to this circle.
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