An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs
type: Item
instance of: scholarly article
title: An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs
zbMATH Open document ID: 1233.65049
author: W. Auzinger
author: Herbert Lehner
author: E. B. Weinmüller
published in: BIT
publication date: 2011-05-04T00:00:00Z
review text: The authors consider linear index-1 differential-algebraic equations (DAEs) with properly stated leading term \[ A(t)(D(t)x(t))' + B(t)x(t)=g(t),\quad t\in (0,1], \tag{1} \] with appropriately smooth data. Further, it is assumed that the inherent ordinary differential equation (ODE) of (1) possesses a singularity of first kind at \(t=0\). The numerical solution of DAEs (1) by stiffly accurate collocation methods is analysed. A posteriori error estimation based on the defect correction is extended from singular ODEs, see \textit{W. Auzinger, O. Koch} and \textit{E. Weinmüller} [Numer. Algorithms 31, No. 1--4, 5--25 (2002; Zbl 1021.65038)], to singular index-1 DAEs (1). As the main result of the paper, using the convergence results for stiffly accurate collocation methods, it is pointed out that the error estimate is asymptotically correct. This means that while the global error of the collocation solution is of order \(h^s\), where \(h\) is the stepsize and \(s\) is the collocation degree, here \(s\) is assumed to be even, the deviation between the global error and the defect-based estimate is of order \(h^{s+1}\). Finally, two numerical examples are given for illustration. In the appendix, the interrelation between collocation methods and implicit Runge-Kutta methods as applied to DAEs is discussed.
reviewed by: Vu Hoang Linh
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L80
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A09
zbMATH DE Number: 5883653
zbMATH Keywords: differential-algebraic equation
zbMATH Keywords: singularity of first kind
zbMATH Keywords: collocation
zbMATH Keywords: a posteriori error estimation
zbMATH Keywords: defect correction
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
zbMATH Keywords: implicit Runge-Kutta methods
describes a project that uses: Sbvp
describes a project that uses: COLNEW
MaRDI profile type: Publication
full work available at URL: s10543-011-0321-9
OpenAlex ID: W2137069878
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DOI: 10.1007/S10543-011-0321-9
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description: scientific article; zbMATH DE number 5883653