Background.

An increasing demand for the simulation of multi-functional circuits is observed in various fields of application (e.g. automotive industry, telecommunication). The design of future chip generations realizing complex system solutions as, e.g., smart power integrated chips causes two conflicting requirements for the simulation: High speed and high reliability. Innovative modeling concepts and a rigorous mathematical analysis are needed for the development of new simulation methods fulfilling both demands.

For circuit design with nanometer scale devices, the inclusion of semiconductor equations models directly into the circuit equations is preferable. Consequently, the resulting model equations represent coupled systems of parabolic PDEs and DAEs (cf. [6]). Benefits from the structure will play a key-role for the development of innovative simulation methods with better performance than already existing mixed circuit/device simulation packages from electrical engineer groups. Furthermore, a mathematical analysis of these coupled systems is missing in the literature so far.

The modeling of such coupled systems is under development in close cooperation with U. Feldmann, D. Estévez Schwarz (Infineon Technologies AG) within the framework of the BMBF project "Schaltungssimulation unter Einbeziehung erweiterter Halbleitermodelle". Furthermore, the results in [2] about the structure and the index of circuit DAEs build the basis for the mathematical and numerical analysis of the coupled systems.

Research program.

This project will be concerned with the numerical analysis of non-stationary coupled systems represented as abstract DAEs with unbounded operators. This class includes the special systems described above. We will focus our work on the necessary mathematical analysis (in particular perturbation analysis) and the determination of suitable discretizations. Recent results on degenerate differential equations in Banach spaces of special canonical form (see [3]), on multi-scaled models of problems of different complexity (cf. [7]) as well as on so called PDAEs (e.g. [1]) shall be considered in conjunction with the classical theory on operator differential equations with bounded operators (e.g. S.G. Kreyn, G.A. Kurina). An approach including an index characterization for abstract DAEs in Hilbert spaces with unbounded operators has recently been developed by the applicants in [4,5].

Cooperation.

We expect the project to benefit from a close cooperation with the projects "Numerical methods for stochastic differential-algebraic equations applied to transient noise analysis in circuit simulation" (R. März, W. Römisch), "Structure analysis for simulation and control problems of differential algebraic equations" (R. März, V. Mehrmann), "Model reduction for large-scale systems in control and circuit simulation" (M. Bollhöfer, V. Mehrmann), "Solution of large unstructured linear systems in circuit simulation" (U. Baur, P. Benner). Furthermore, we plan to contribute to the projects "Current mathematics at school" (J. Kramer) and "Teachers at universities" (J. Kramer) by special talks and seminars for pupils and teachers.

Literatur.