# John C. Butcher

This paper presents a review of the role played by trees in the theory of Runge–Kutta methods. The use of trees is in contrast to early publications on numerical methods, in which a deceptively simpler approach was used. This earlier approach is not only non-rigorous, but also incorrect. It is now known, for example, that methods can have different orders(More)
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• General linear methods for ordinary differential equations – p. 1/46 Solving ordinary differential equations numerically is, even today, still a great challenge. This applies especially to stiff differential equations and to closely related problems involving algebraic constraints (DAEs). Although the problem seems to be solved — there are already highly(More)
Ab.stract. it has been sho~v~, by I)ahlquist [1] that , q~ k step method for the numericet 15 soll~tion of an orditlary differential equation is unstable unless the order is less than /:93 This paper is concerned with a modificati(m t~o the form of the multistep process such ti~r higher orders can be ~ttained. For k.~7 examples of such modified processes of(More)
• 1997
A new representation for diagonally implicit multistage integration methods (DIMSIMs) is derived in which the vector of external stages directly approximates the Nordsieck vector. The methods in this formulation are zero-stable for any choice of variable mesh. They are also easy to implement since changing step-size corresponds to a simple rescaling of the(More)
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• 2004
We describe the construction of explicit general linear methods of order p and stage order q=p with s=p+1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends on one(More)
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• 2005
General linear methods are extended to the case in which second derivatives, as well as first derivatives, can be calculated. Methods are constructed of third and fourth order which are A-stable, possess the Runge–Kutta stability property and have a diagonally implicit structure for efficient implementation.
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• 2014
This version is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. G-symplectic general linear methods are designed to approximately preserve sym-plectic invariants for Hamiltonian systems. In this paper, the properties of G-symplectic methods are explored computationally and(More)
BACKGROUND A retrospective matched (1:1) case-control study was conducted to compare the financial impact and costs attributable to ventilator-associated pneumonia (VAP) in a 25-bed pediatric intensive care unit (PICU) in a 475-bed quaternary-care pediatric hospital from the perspective of multiple stakeholders, including the hospital and payors. METHODS(More)
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