Numerical methods for ordinary differential equations in the 20th century
type: Item
instance of: scholarly article
title: Numerical methods for ordinary differential equations in the 20th century
zbMATH Open document ID: 0969.65063
DOI: 10.1016/S0377-0427(00)00455-6
published in: Journal of Computational and Applied Mathematics
publication date: 2001-09-25T00:00:00Z
review text: The paper is a survey following many of the main strands in the development of numerical methods for general initial value problems, stiff systems, or special problem types. The early contributions of Bashforth, Adams and Runge, together with an introduction to the fundamental work of Euler form the subject of a first section. Further sections deal either with specific periods of time or with contributions with a unifying scheme: Heun, Nyström and Moulton papers, Milne's device, Taylor series, modern theory of linear multistep methods and Runge-Kutta methods. Nontraditional methods are also revised. A special section is dedicated to stiff problems, and another to the beginnings of differential equation software. The last section treats some special problems.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 65-03
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1565999
zbMATH Keywords: initial value problems
zbMATH Keywords: linear multistep methods
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: stiff problems
zbMATH Keywords: historical survey
zbMATH Keywords: Adams-Bashforth method
zbMATH Keywords: Euler method
zbMATH Keywords: Heun method
zbMATH Keywords: Nyström method
zbMATH Keywords: Adams-Moulton method
zbMATH Keywords: Milne method
author: J. C. Butcher
reviewed by: Dana Petcu
describes a project that uses: DIFSUB
describes a project that uses: DASSL
MaRDI profile type: Publication
cites work: A New Theoretical Approach to Runge–Kutta Methods
cites work: Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s
cites work: Brèves communications. Deux théorèmes sur la $A-$stabilité des schémas de Runge-Kutta simplement implicités
cites work: The automatic solution of systems of ordinary differential equations by the method of Taylor series
cites work: Numerical treatment of ordinary differential equations by extrapolation methods
cites work: A special family of Runge-Kutta methods for solving stiff differential equations
cites work: Stability Criteria for Implicit Runge–Kutta Methods
cites work: Coefficients for the study of Runge-Kutta integration processes
cites work: Implicit Runge-Kutta Processes
cites work: On the Attainable Order of Runge-Kutta Methods
cites work: A Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations
cites work: An Algebraic Theory of Integration Methods
cites work: The non-existence of ten stage eighth order explicit Runge-Kutta methods
cites work: A stability property of implicit Runge-Kutta methods
cites work: Q5332835
cites work: Q4113355
cites work: Sur la B-stabilité des méthodes de Runge-Kutta
cites work: Integration of Stiff Equations
cites work: Convergence and stability in the numerical integration of ordinary differential equations
cites work: Q3256529
cites work: A special stability problem for linear multistep methods
cites work: Q4122713
cites work: G-stability is equivalent toA-stability
cites work: Cyclic Composite Multistep Predictor-Corrector Methods
cites work: A-Stable Methods and Padé Approximations to the Exponential
cites work: Comparing numerical methods for stiff systems of O.D.E:s
cites work: The Concept of B-Convergence
cites work: Hybrid Methods for Initial Value Problems in Ordinary Differential Equations
cites work: A Program for the Automatic Integration of Differential Equations using the Method of Taylor Series
cites work: Q5803623
cites work: Generalized Multistep Predictor-Corrector Methods
cites work: The numerical solution of differential-algebraic systems by Runge-Kutta methods
cites work: On the Butcher group and general multi-value methods
cites work: A theory for Nyström methods
cites work: Q5719651
cites work: Comparing Numerical Methods for Ordinary Differential Equations
cites work: Q4000414
cites work: A study of Rosenbrock-type methods of high order
cites work: Q3241447
cites work: Q5600158
cites work: On Numerical Integration of Ordinary Differential Equations
cites work: One-step methods of hermite type for numerical integration of stiff systems
cites work: Order conditions for Rosenbrock type methods
cites work: On the Stability and Accuracy of One-Step Methods for Solving Stiff Systems of Ordinary Differential Equations
cites work: Some general implicit processes for the numerical solution of differential equations
cites work: Q4358703
cites work: Block Implicit One-Step Methods
cites work: Order stars and stability theorems

Numerical methods for ordinary differential equations in the 20th century (full work available at URL)
full work available at URL: s0377-0427(00)00455-6

Numerical methods for ordinary differential equations in the 20th century (OpenAlex ID)
OpenAlex ID: W2065960630

Numerical methods for ordinary differential equations in the 20th century (Wikidata QID)
Wikidata QID: Q127982900