The choice of parameters in parallel general linear methods for stiff problems
type: Item
instance of: scholarly article
title: The choice of parameters in parallel general linear methods for stiff problems
zbMATH Open document ID: 0949.65076
DOI: 10.1016/S0168-9274(99)00035-5
author: A. D. Singh
author: J. C. Butcher
published in: Applied Numerical Mathematics
publication date: 2000-11-21T00:00:00Z
review text: This paper studies the possibilities of some diagonally implicit multistage integration (DIMSI) methods for the numerical solution of stiff initial value problems in a parallel environment. The methods under consideration, which are a subclass of general linear methods, were proposed by the first author [Appl. Numer. Math. 11, No. 5, 347-363 (1993; Zbl 0773.65046)] in a more general setting. Here the authors discuss some particular choices of the available parameters in these methods taking into account not only the standard stability and order properties but also the implementation details. More specifically, a DIMSI method defined by the matrices \((A,U,B,V)\) advances the numerical solution from \(t_n\) to \(t_{n+1}=t_n+h\) by means of a formula of type \(y^{[n+1]}= V y^{[n]}+ h B F(Y)\) where \(y^{[n]}\) is the external stage vector at time \(t_n\) which collects the numerical information on the solution at time \(t_n\) (typically this information is given by a Nordsieck vector \((y_n, h y_n'', \ldots, h^p y_n^{(p)}))\) and \(Y\) is the internal stage vector that satisfies the algebraic equation \(Y = h A F(Y)+ U y^{[n]}\). In the first part of the paper the authors assume \( A = \lambda I\) and \(V\) a rank one matrix. In this context they derive a family of A-stable methods with different orders and some available parameters, but its main drawback is that they possess large error constants. In the second part of the paper \(A\) is assumed to be a diagonal matrix with non equal diagonal elements and \(U\) a rank two matrix. In this more general setting several methods with good stability properties and smaller error constants are derived. Finally the results of some numerical experiments are presented to compare the behaviour of the new (parallel) methods with other standard sequential solvers with some test problems.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65Y05
zbMATH DE Number: 1455716
zbMATH Keywords: diagonally implicit multistage integration methods
zbMATH Keywords: linear methods
zbMATH Keywords: stiff problems
zbMATH Keywords: parallel computation
zbMATH Keywords: error bounds
zbMATH Keywords: stability
zbMATH Keywords: order
zbMATH Keywords: numerical experiments
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: s0168-9274(99)00035-5
OpenAlex ID: W2059378532
description: scientific article; zbMATH DE number 1455716