Michael Günther

Multirate Partitioned Runge-Kutta Methods

The coupling of subsystems in a hierarchical modelling approach leads to different time constants in the dynamical simulation of technical systems. Multirate schemes exploit the different time scales by using different time steps for the subsystems. The stiffness of the system or at least of some subsystems in chemical reaction kinetics or network analysis, for example, forbids the use of explicit integration schemes. To cope with stiff problems, we introduce multirate schemes based on partitioned Runge—Kutta methods which avoid the coupling between active and latent components based on interpolating and extrapolating state variables. Order conditions and test results for such a lower order MPRK method are presented.

Preconditioned Dynamic Iteration for Coupled Differential-Algebraic Systems

The network approach to the modelling of complex technical systems results frequently in a set of differential-algebraic systems that are connected by coupling conditions. A common approach to the numerical solution of such coupled problems is based on the coupling of standard time integration methods for the subsystems. As a unified framework for the convergence analysis of such multi-rate, multi-method or dynamic iteration approaches we study in the present paper the convergence of a dynamic iteration method with a (small) finite number of iteration steps in each window. Preconditioning is used to guarantee stability of the coupled numerical methods. The theoretical results are applied to quasilinear problems from electrical circuit simulation and to index-3 systems arising in multibody dynamics.

The DAE-index in electric circuit simulation

The index of differential-algebraic equations is one measure for the numerical problems in electric circuit simulation. The index depends on the mathematical model in different ways: setup of equations, classical or charge-oriented formulation and modelling of basic elements and semiconductor devices. The modelling of MOS transistor circuits will show this in more detail.

A multirate W-method for electrical networks in state-space formulation

Subunits of coupled technical systems typically behave on differing time scales, which are often separated by several orders of magnitude. An ordinary integration scheme is limited by the fastest changing component, whereas so-called multirate methods employ an inherent step size for each subsystem to exploit these settings. However, the realization of the coupling terms is crucial for any convergence. Thus the approach to return to one-step methods within the multirate concept is promising. This paper introduces the multirate W-method for ordinary differential equations and gives a theoretical discussion in the context of partitioned Rosenbrock-Wanner methods. Finally, the MATLAB implementation of an embedded scheme of order (3)2 is tested for a multirate version of Prothero-Robinsons equation and the inverter-chain-benchmark.

Dynamic Iteration for Coupled Problems of Electric Circuits and Distributed Devices

Coupled systems of differential-algebraic equations (DAEs) may suffer from instabilities during a dynamic iteration. We extend the existing analysis on recursion estimates, error propagation, and stability to (semiexplicit) index-1 DAEs. In this context, we discuss the influence of certain coupling structures and the computational sequence of the subsystems on the rate of convergence. Furthermore, we investigate in detail convergence and divergence for two coupled problems stemming from refined electric circuit simulation. These are the semiconductor-circuit and field-circuit couplings. We quantify the convergence rate and behavior also using Lipschitz constants and suggest an enhanced modeling of the coupling interface in order to improve convergence.

Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction

In this paper we present a novel highly efficient approach to determine material properties from measurement results. We apply our method to thermal properties of thin-film multilayers with three different materials, amorphous silicon, silicon nitride and silicon oxide. The individual material properties are identified by solving an optimization problem. For this purpose, we build a parameterized reduced-order model from a finite element (FE) model and fit it to the measurement results. The use of parameterized reduced-order models within the optimization iterations speeds up the transient solution time by several orders of magnitude, while retaining almost the same precision as the full-scale model.

Parabolic Differential-Algebraic Models in Electrical Network Design

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multidimensional approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that joins the differential-algebraic initial value problem for the electric circuit with parabolic-elliptic boundary value problems modeling the diodes. For this mixed initial boundary value problem of partial differential-algebraic equations a first existence and uniqueness result is given, and its asymptotic behavior is discussed.

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

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.

A PDAE Model for Interconnected Linear RLC Networks

In electrical circuit simulation, a refined generalized network approach is used to describe secondary and parasitic effects of interconnected networks. Restricting our investigations to linear RLC circuits, this ansatz yields linear initial-boundary value problems of mixed partial-differential and differential-algebraic equations, so-called PDAE systems. If the network fulfils some topological conditions, this system is well-posed and has perturbation index 1 only: the solution of a slightly perturbed system does not depend on derivatives of the perturbations. As method-of-lines applications are often used to embed PDAE models into time-domain network analysis packages, it is reasonable to demand that the analytical properties of the approximate DAE system obtained after semidiscretization are consistent with the original PDAE system. Especially, both should show the same sensitivity with respect to initial and boundary data. We will learn, however, that semidiscretization may act like a deregularization of an index-1 PDAE model, if an inappropriate type of semidiscretization is used.

A Generalized-Structure Approach to Additive Runge--Kutta Methods

This work considers a general structure of the additively partitioned Runge--Kutta methods by allowing for different stage values as arguments of different components of the right-hand side. An order conditions theory is developed for the new formulation of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit and implicit-implicit methods in the new framework. The new approach, named GARK, introduces additional flexibility when compared to traditional partitioned Runge--Kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.