On implicit Euler for high-order high-index DAEs
type: Item
instance of: scholarly article
title: On implicit Euler for high-order high-index DAEs
zbMATH Open document ID: 1005.65081
DOI: 10.1016/S0168-9274(01)00164-7
author: R. Smith
published in: Applied Numerical Mathematics
publication date: 2002-08-22T00:00:00Z
review text: The present paper deals with high order differential algebraic equations of the form \[ \begin{aligned} y^{(d)} &= f(t,y,y^{(1)},\dots, y^{(d-1)}, \lambda),\;y^{(j)}(t_0)= \eta_j,\;j= 0,1,\dots, d-1,\\ 0 &= g(t,y,y^{(1)},\dots, y^{(d-1)}),\;r\in [2,d+1].\end{aligned}\tag{1} \] It is well known that the application of the implicit Euler method to (1) in general fails to converge, since the accuracy of the \(y(t_n)\) estimate is to low to produce reasonable estimates of the derivatives of \(y\). As a remedy a slight modification of the implicit Euler method is suggested. Under appropriate assumptions uniform convergence is shown in the case \(d= r+1\). Moreover, generalizations of that approach to backward differentiation formulae are presented.
Mathematics Subject Classification ID: 65L80
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 1787745
zbMATH Keywords: high order differential algebraic equations
zbMATH Keywords: implicit Euler method
zbMATH Keywords: convergence
zbMATH Keywords: backward differentiation formulae
reviewed by: Johannes Schropp
describes a project that uses: LSODE
describes a project that uses: DASSL
MaRDI profile type: Publication
cites work: Improving the accuracy of BDF methods for index 3 differential-algebraic equations
cites work: Developing software for time-dependent problems using the method of lines and differential-algebraic integrators
cites work: Q4868578
cites work: Backward Differentiation Approximations of Nonlinear Differential/Algebraic Systems
cites work: Q3957868
cites work: Q3659053
cites work: Convergence of a class of Runge-Kutta methods for differential-algebraic systems of index 2
full work available at URL: s0168-9274(01)00164-7
OpenAlex ID: W2028759922
description: scientific article; zbMATH DE number 1787745