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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 3/2017

07.04.2017

Conversion methods for improving structural analysis of differential-algebraic equation systems

verfasst von: Guangning Tan, Nedialko S. Nedialkov, John D. Pryce

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2017

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Abstract

Structural analysis (SA) of a system of differential-algebraic equations (DAEs) is used to determine its index and which equations to be differentiated and how many times. Both Pantelides’s algorithm and Pryce’s \(\varSigma \)-method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates \(\varSigma \)-method’s failures and presents two conversion methods for fixing them. Under certain conditions, both methods reformulate a DAE system on which the \(\varSigma \)-method fails into a locally equivalent problem on which SA is more likely to succeed. Aiming at achieving global equivalence between the original DAE system and the converted one, we provide a rationale for choosing a conversion from the applicable ones.
Vorheriger Artikel Stability radii for real linear Hamiltonian systems with perturbed dissipation
Nächster Artikel Symbols and exact regularity of symmetric pseudo-splines of any arity

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Fußnoten
1
The colon notation \(p\,{:}\,q\) for integers p, q denotes either the unordered set or the enumerated list of integers i with \(p\le i\le q\), depending on context.
 
2
Throughout this article, “derivatives of \(x_j\)” include \(x_j\) itself as its 0th derivative: \(x_j^{(l)}=x_j\) if \(l=0\).
 
3
When we present a DAE example, we also present its signature matrix \(\varSigma \), the canonical offset pair \((\mathbf {c};\mathbf {d})\), and the associated System Jacobian \(\mathbf {J}\).
 
4
We consider it with parameters \(\beta =\epsilon =1\), \(\alpha _1=\alpha _2=\delta =1\), and \(\gamma =-1\), and we use subscripts for parameter indices. The equations \(g_1,g_2\) are renamed \(f_3, f_4\) and the variables \(y_1,y_2\) are renamed \(x_3, x_4\).
 
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Metadaten
Titel
Conversion methods for improving structural analysis of differential-algebraic equation systems
verfasst von
Guangning Tan
Nedialko S. Nedialkov
John D. Pryce
Publikationsdatum
07.04.2017
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 3/2017
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-017-0655-z

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