On the Tractability Index of a Class of Partial Differential-Algebraic Equations

Alì, Giuseppe; Rotundo, Nella

doi:10.1007/s10440-012-9722-2

Cited by 5 publications

(9 citation statements)

References 13 publications

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“…Next, we wish to find additional conditions, depending on the coupling matrices U , V , such that this decoupling is also preserved for the nonlinear equation (3.2). A general discussion can be found in [3]. In the following, we derive appropriate additional conditions for index-1 and index-2 matrix pencils with a nonlinear coupling term.…”

Section: Basics Of the Tractability Index And Related Projectorsmentioning

confidence: 99%

“…It is possible to prove that the additional condition (3.10), together with the index-1 conditions for the matrix pencil A − λE, imply that (3.7), with algebraic , is an index-1 system [3]. In general, condition (3.11), or, equivalently, condition (3.12), is different from (3.10), even when U = V , unless we can choose Q 0 = Q 0 .…”

Section: Index-1 Matrix Pencil a − λEmentioning

confidence: 99%

“…Moreover, if is a nonlinear function and A −λE is an index-2 matrix pencil, then the additional condition (3.18) implies that (3.17) is an index-2 system according to the definition given in [11] (see [3]). Proposition 2 below explores the topological implications of condition (3.19).…”

Section: Index-2 Matrix Pencil a − λEmentioning

confidence: 99%

“…For the extension of the index concept to these kinds of equations, we follow the approach introduced in [3]. The idea is to determine additional topological conditions on the coupling matrices which relate the DAE and PDE parts, such that the DAE index keeps its validity for the enlarged system.…”

Section: Introductionmentioning

confidence: 99%

“…It might be possible to adopt the quasi-Weierstrass approach proposed in [4], but this strategy is not pursued here.…”

Section: Introductionmentioning

confidence: 99%

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Index-2 elliptic partial differential-algebraic models for circuits and devices

Alì

Bartel

Rotundo

2015

Journal of Mathematical Analysis and Applications

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We consider an elliptic partial differential-algebraic model which arises in the modeling of an electric network that contains semiconductor devices. In this context, the electric network is described by linear differential-algebraic equations, while the semiconductor devices are described by nonlinear elliptic partial differential equations. The coupling takes place through the source term for the network equations and the boundary conditions for the device equations. Under the assumption that the fully coupled model has tractability index 2, we prove an existence result for this system. The extension of the notion of tractability index to a specific class of coupled systems is a further result of this paper.

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Section: Basics Of the Tractability Index And Related Projectorsmentioning

confidence: 99%

Section: Index-1 Matrix Pencil a − λEmentioning

confidence: 99%

Section: Index-2 Matrix Pencil a − λEmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It might be possible to adopt the quasi-Weierstrass approach proposed in [4], but this strategy is not pursued here.…”

Section: Introductionmentioning

confidence: 99%

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Index-2 elliptic partial differential-algebraic models for circuits and devices

Alì

Bartel

Rotundo

2015

Journal of Mathematical Analysis and Applications

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Existence and uniqueness of solution for multidimensional parabolic partial differential‐algebraic equations arising in semiconductor modeling

Alì

Rotundo

2021

Math Methods in App Sciences

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This paper concerns with a compact network model combined with distributed models for semiconductor devices. For linear RLC networks containing distributed semiconductor devices, we construct a mathematical model that joins the differential‐algebraic initial value problem for the electric circuit with multidimensional parabolic‐elliptic boundary value problems for the devices. We prove an existence and uniqueness result and the asymptotic behavior of this mixed initial boundary value problem of partial differential‐algebraic equations.

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Index-Aware Model-Order Reduction for a Special Class of Nonlinear Differential-Algebraic Equations

et al. 2021

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We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term $$\mathbf{f}(\mathbf{E}\mathbf{x}),$$ f ( E x ) , where $$\mathbf{E}$$ E is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and system size. Model-order reduction aims to eliminate this problem by generating reduced-order models that have lower computational cost to simulate, yet accurately represent the original large-scale system behavior. However, direct reduction and simulation of nonlinear differential-algebraic equations is difficult due to hidden constraints which affect the choice of numerical integration methods and model-order reduction techniques. We propose an extension of index-aware model-order reduction methods to a special class of nonlinear differential-algebraic equations without any kind of linearization. The proposed model-order reduction approach involves automatic decoupling of nonlinear differential-algebraic equations into nonlinear ordinary differential equations and algebraic equations, based on the decoupling of the linear differential equations obtained by ignoring the nonlinear term, thanks to an additional structural condition. This allows applying standard model-order reduction techniques to both parts without worrying about the index. The same procedure can also be used to simulate nonlinear differential-algebraic equations by standard integration schemes.

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On the Tractability Index of a Class of Partial Differential-Algebraic Equations

Cited by 5 publications

References 13 publications

Index-2 elliptic partial differential-algebraic models for circuits and devices

Index-2 elliptic partial differential-algebraic models for circuits and devices

Existence and uniqueness of solution for multidimensional parabolic partial differential‐algebraic equations arising in semiconductor modeling

Index-Aware Model-Order Reduction for a Special Class of Nonlinear Differential-Algebraic Equations

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