Initial value problems: Numerical methods and mathematics
type: Item
instance of: scholarly article
title: Initial value problems: Numerical methods and mathematics
zbMATH Open document ID: 0817.65059
DOI: 10.1016/0898-1221(94)00182-0
publication date: 1994-12-01T00:00:00Z
review text: The author summarizes his research on Runge-Kutta (RK) and related general linear methods for approximation to the solution of ordinary differential equations. An explanation is given of the way to apply a general series, now called a B-series (Butcher series), to the study of particular RK algorithms. A proof is given of the A-stability of implicit RK algorithms. Criteria for stability and consistency of general linear algorithms are presented. Finally a criterion is given for the stability of a new class of algorithms, called diagonally implicit multistage integration methods (DIMSIM's), developed by the author. Some of these are designed to treat stiff systems by means of parallel computations.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E13
zbMATH DE Number: 699063
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: Butcher series
zbMATH Keywords: general linear methods
zbMATH Keywords: B-series
zbMATH Keywords: A- stability
zbMATH Keywords: consistency
zbMATH Keywords: diagonally implicit multistage integration methods
zbMATH Keywords: stiff systems
zbMATH Keywords: parallel computations
author: J. C. Butcher
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
full work available at URL: 0898-1221(94)00182-0
OpenAlex ID: W2078033749
cites work: Q4725642
cites work: Implicit Runge-Kutta Processes
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cites work: Non-linear stability of a general class of differential equation methods
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cites work: OnA-stable implicit Runge-Kutta methods
cites work: On the implementation of implicit Runge-Kutta methods
cites work: A Transformed implicit Runge-Kutta Method
cites work: The Order of Numerical Methods for Ordinary Differential Equations
cites work: Q5577991
cites work: Diagonally-implicit multi-stage integration methods
cites work: Q4860568
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 699063