Numerical integration of positive linear differential-algebraic systems

Baum, Ann-Kristin; Mehrmann, Volker

doi:10.1007/s00211-013-0514-z

Cited by 5 publications

(10 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The matrix T = [T 1 , T 2 ] ∈ R n×n contains a basis of the generalized eigenspaces geig reg (E) := ∪ λ∈σ(E) geig(λ) and geig 0 (E), respectively. Noting that 6) we find that ker(E ν ) = eig 0 (E) and range(E ν ) = geig reg (E).…”

Section: Generalized Inverses 15mentioning

confidence: 59%

“…Construction of a discrete flow-on-manifold formulation and its application to positive systems 215 Theorem 6.2.1 combines the results of Theorem 4 and 6 given in [6]. Theorem 4 ensures a nonnegative discretization of the ODEẋ d =Ê DÂ x d +Ê Db obtained via the Drazin projection, cp.…”

Section: Summary Positivity Conceptsmentioning

confidence: 98%

“…(4.122), the preservation of positivity has been analyzed in [6]. For (6.16) With these notations, in [6] we have specified the positivity threshold for the method (A, β, γ).…”

Section: Summary Positivity Conceptsmentioning

confidence: 99%

“…(4.122), the preservation of positivity has been analyzed in [6]. For (6.16) With these notations, in [6] we have specified the positivity threshold for the method (A, β, γ). where R (A,β,γ) and Q (A,β,γ) are the monotonicity radii of the stability functions φ(z) := 1 + zβ T (I s − zA) −1 e and η(z) := β T (I s − zA) −1 e i for i = 1, ..., n.…”

Section: Summary Positivity Conceptsmentioning

confidence: 99%

“…For linear systems with constant coefficients, positivity has been characterized in [137]. Conditions under which numerical solutions of these systems inherit the property of positivity, have been presented in [6]. The optimal control of linear systems with time-varying coefficients has been addressed in [40].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

Baum

2015

J Dyn Diff Equat

View full text Add to dashboard Cite

Differential-algebraic equations (DAEs) are coupled systems of differential and algebraic equations arising in the modeling of dynamic processes that are restricted by auxiliary, algebraic constraints. Examples are, e.g., multibody systems, where the movement of the system is forced to a predefined trajectory by connected joints, electrical circuits or networks, where connections and loops lead to algebraic relations of the flow and effort variables, or biological systems, reaction networks or advection-diffusion equations, where the evolution of the process is restricted by balance equations and conservation laws. Modeling dynamic processes in economic or social sciences, in biological or chemical engineering, the analyzed values typically represent real-valued quantities like the amount of goods or individuals or the density of a chemical or biological species that cannot take negative values. In combination with the algebraic constraints, this property leads to the notion of positive DAEs, i.e., systems whose solutions remain componentwise nonnegative whenever the initial value is nonnegative.In this work, we generalize the concept of the flow from ordinary differential equations (ODEs) to DAEs and use this framework to study system properties like invariant sets and positivity. To decouple the differential and algebraic components in a DAE, we develop an approach that allows to parameterize embedded submanifolds using projections and that extends the idea of projections to embedded submanifold. Combined with the theory of the strangeness-index, we apply this projection approach to remodel a given DAE as a set of explicit differential and algebraic equations. Solving the decoupled system and showing existence and uniqueness of the solution representation, we generalize the notion of the flow to DAEs. For linear systems, we generalize Duhamel's formula. Based on the flow, we extend the results of invariant sets from ODEs to DAEs and we characterize linear and nonlinear DAEs with regard to positivity. ZUSAMMENFASSUNG

show abstract

Section: Generalized Inverses 15mentioning

confidence: 59%

Section: Summary Positivity Conceptsmentioning

confidence: 98%

“…(4.122), the preservation of positivity has been analyzed in [6]. For (6.16) With these notations, in [6] we have specified the positivity threshold for the method (A, β, γ).…”

Section: Summary Positivity Conceptsmentioning

confidence: 99%

Section: Summary Positivity Conceptsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

Baum

2015

J Dyn Diff Equat

View full text Add to dashboard Cite

show abstract

A Symbolic–Numerical Method for Integration of DAEs Based on Geometric Control Theory

Freitas

Silva²

2014

J Control Autom Electr Syst

View full text Add to dashboard Cite

On finite element method–flux corrected transport stabilization for advection–diffusion problems in a partial differential–algebraic framework

Vuong

Simeon

2014

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

An extension of the finite element method-flux corrected transport stabilization (FEM-FCT) for hyperbolic problems in the context of partial differentialalgebraic equations (PDAEs) is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ−scheme when applied to the time integration of the resulting differentialalgebraic equation (DAE) is shown, under a mild restriction on the time stepsize. As crucial tool in the analysis, the Drazin inverse and the corresponding Drazin ODE are explicitly derived. Numerical results are presented for nonconstant and time-dependent boundary conditions in one space dimension and for a two-dimensional advection problem where the advection proceeds skew to the mesh.

show abstract

Numerical integration of positive linear differential-algebraic systems

Cited by 5 publications

References 33 publications

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

A Symbolic–Numerical Method for Integration of DAEs Based on Geometric Control Theory

On finite element method–flux corrected transport stabilization for advection–diffusion problems in a partial differential–algebraic framework

Contact Info

Product

Resources

About