Preprint 02-2012

Numerical Integration of Positive Linear Differential-Algebraic Systems

Author(s) : Ann-Kristin Baum , Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 02-2012

MSC 2000

65L80 Methods for differential-algebraic equations

65L06 Multistep, Runge-Kutta and extrapolation methods

Abstract :
In the simulation of dynamical processes in economy, social sciences, biology or chem- istry, 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 de- scribed 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 al- gebraic 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.

Keywords : Differential-algebraic equation, positive system, Runge-Kutta method, multistep method, positivity preserving discretization, Z-matrix, M-matrix, stability function, Drazin inverse