Index Concepts for Linear Mixed Systems of Differential-Algebraic and Hyperbolic-Type Equations

Abstract

For many technical systems, the use of a refined network modeling approach leads to hyperbolic-type initial-boundary value problems of partial differential-algebraic equations (PDAEs). The boundary conditions of these systems are governed by time-dependent differential-algebraic equations (DAEs) that couple the PDAE system with the network elements that are modeled by DAEs in time only. In order to classify these systems, we extend some index notions that have already been introduced to treat parabolic-type problems. A perturbation index is considered that reflects the sensitivity of the mixed system to slight perturbations in the right-hand side of the PDAEs, as well as in the input signals of the DAE systems and initial values. In order to make an a posteriori analysis of semidiscretization in space and time, we introduce additionally a space and a method of lines (MOL) index. Here one is especially interested in whether the semidiscretized systems properly reflect the properties of the underlying systems. We will show that these indices may detect an artificial sensitivity with respect to perturbations, e.g., if the semidiscretization does not consider the information transport along the characteristics.