A proposed numerical algorithm for solving nonlinear index problems

Chung, Yonsoo; Westerberg, Arthur W.

doi:10.1021/ie00103a023

Cited by 23 publications

(17 citation statements)

References 10 publications

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“…The concept of index relates to several questions in systems and control theory and is of active interest in the engineering literature [Bachman, Brüll, Mrziglod, and Pallaske (1990), Blajer (1992), Chung and Westerberg (1990), Dai (1989), Fliess, Lévine and Rouchon (1993a), Lefkopoulos and Stadtherr (1993)]. The typical starting point in many nonlinear control problems is a system…”

Section: Systems and Control Theorymentioning

confidence: 99%

The index of general nonlinear DAEs

Campbell

Gear

1995

Numerische Mathematik

312

157

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In the last few years there has been considerable research on differential algebraic equations (DAEs) F (t, y, y ) = 0 where F y is identically singular. Much of the mathematical effort has focused on computing a solution that is assumed to exist. More recently there has been some discussion of solvability of DAEs. There has historically been some imprecision in the use of the two key concepts of solvability and index for DAEs. The index is also important in control and systems theory but with different terminology. The consideration of increasingly complex nonlinear DAEs makes a clear and correct development necessary. This paper will try to clarify several points concerning the index. After establishing some new and more precise terminology that we need, some inaccuracies in the literature will be corrected. The two types of indices most frequently used, the differentiation index and the perturbation index, are defined with respect to solutions of unperturbed problems. Examples are given to show that these indices can be very different for the same problem. We define new "maximum indices, " which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other. These indices are also related to an index defined in terms of Jacobians. (1991): 65L20 Mathematics Subject Classification

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Section: Systems and Control Theorymentioning

confidence: 99%

The index of general nonlinear DAEs

Campbell

Gear

1995

Numerische Mathematik

312

157

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“…To overcome these difficulties, techniques involving combination of algebraic manipulations and differentiations have been proposed to reduce high-index DAEs to ODES (Gear and Petzold, 1984;Gear, 1988) or index-one DAEs (Bachmann et al, 1990). These index reduction techniques have been used as the basis for the majority of proposed numerical simulation methods for high-index DAE systems (Gear and Petzold, 1984;Petzold, 1986;Chung and Westerberg, 1990;Secchi et al, 1993). Nonlinear constrained optimization techniques have also been proposed for this purpose (Renfro et al, 1987;Jarvis and Pantelides, 1992).…”

Section: Introductionmentioning

confidence: 99%

Feedback control of nonlinear differential‐algebraic‐equation systems

1995

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“…The numerical singularry of Q'• is a concern for large multidimensional calculations as the system evolves in time and space. If the numerical and/or structural singularities present in higher-index DAE systems can be automatically detected [Chung and Westerberg, 1990], then an index reduction scheme may be used locally to reformulate the equations to have a lower index. Currently, however, this is beyond the capabilities of most software for groundwater chemistry and transport.…”

Section: Singular Q'•mentioning

confidence: 99%

An analysis of complex reaction networks in groundwater modeling

Chilakapati¹,

Ginn²,

Szecsody³

1998

Water Resources Research

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Abstract. The complex chemistry describing the biogeochemical dynamics in the natural subsurface environments gives rise to heterogeneous reaction networks, the individual segments of which can feature a wide range of timescales. This paper presents a formulation of the mass balance equations for the batch chemistry and the transport of groundwater contaminants participating in such arbitrarily complex networks of reactions. We formulate the batch problem as an initial-value differential algebraic equation (DAE) system and compute its "index" so that the ease of solvability of the system is determined. We show that when the equilibrium reactions obey the law of mass action, the index of this initial-value DAE system is always unity (thus solvable with well-developed techniques) and that the system can be decoupled into a set of linearly implicit ordinary differential equations and a set of explicit algebraic equations. The formulations for the transport of these reaction networks can take advantage of their solvability properties under batch conditions. To avoid the error associated with time splitting fast reactions from transport, we present a split-kinetics approach where the fast equilibrium reactions are combined with transport equations while only the slower kinetic reactions are time split. These results are used to formulate and solve a simplified reaction network for the biogeochemical transformation of Co(II) ethylenediaminetetraacetic acid (EDTA) in the presence of iron-coated sediments.

show abstract

A proposed numerical algorithm for solving nonlinear index problems

Cited by 23 publications

References 10 publications

The index of general nonlinear DAEs

The index of general nonlinear DAEs

Feedback control of nonlinear differential‐algebraic‐equation systems

An analysis of complex reaction networks in groundwater modeling

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