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Main Title: Numerical Integration of Positive Linear Differential-Algebraic Systems
Author(s): Baum, Ann-Kristin
Mehrmann, Volker
Type: Research Paper
Abstract: In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent nonnegative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge-Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We test these conditions for some common Runge-Kutta and multistep methods and observe that none of these methods is suitable to solve positive higher index DAEs in a proper way.
Subject(s): differential-algebraic equation
positive system
Runge-Kutta method
multistep method
positivity preserving discretization
stability function
Drazin inverse
Issue Date: 22-Jan-2012
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 65L80 Methods for differential-algebraic equations
65L06 Multistep, Runge-Kutta and extrapolation methods
15A16 Matrix exponential and similar functions of matrices
15B48 Positive matrices and their generalizations; cones of matrices
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2012, 02
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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