Ulrich Nowak

Massively Parallel Linearly-Implicit Extrapolation Algorithms as a Powerful Tool in Process Simulation*

We study the parallelization of linearly-implicit extrapolation codes for the solution of large scale PDE systems and differential algebraic equations on distributed memory machines. The main advantage of these algorithms is that they enable adapativity both in time and space. Additive Krylov-Schwarz methods yield high parallel perfomance for extrapolation methods. Our approach combines a slightly overlapping domain decomposition together with a polynomial block Neumann preconditioner and a reduced system technique. A further speedup we got by the explicit computation of the matrix-products of the preconditioner and the matrix of the linear system. The parallel algorithms exhibit scalability up to 64 processors already for medium-sized test problems. We show that the codes are really efficient in large application systems for chemical engineering problems.

Recent Developments in Chemical Computing

Abstract This paper surveys three aspects of chemical computing, which seem to play a role in recent developments. First, extrapolation methods for the numerical treatment of differential-algebraic equations are introduced. The associated extrapolation code LIMEX has reached a certain level of sophistication, which makes it a real competitor to the elsewhere widely used multi-step code DASSL of Petzold. Second, adaptive method of lines for partial differential equations such as those arising in combustion problems are treated. Both static and dynamic regridding techniques are discussed in some detail. Finally, some new ideas about the treatment of the kinetic equations arising from polymer reactions are presented. The new feature of the suggested approach is the application of a Galerkin procedure using sets of orthogonal polynomials over a discrete variable (which, of course, in the case of polymer reactions is the polymer degree). The new approach may open the door to a new reliable low-dimensional treatment of complex polymer reactions.

Adaptive Algorithms in Dynamical Process Simulation

Dynamical process simulation plays an increasingly important role in the design and control of chemical plants. Mathematically speaking, the simulation of such processes requires the numerical solution of systems of partial differential equations (PDEs) of reaction-diffusion type with possibly mild convection. In contrast to some other fields of application, time dependence of the process is of real interest. Moreover, due to additional constraints, differential-algebraic equations will naturally arise. As for the spatial geometry, radial or simply plane symmetry will result in 1-D problems, whereas more complex situations will lead to 2-D or even 3-D models, often only given in the form of some CAD input. Even though such problems have been around for quite a while, they still represent a class of hard problems. For this reason, the development of robust and fast algorithms has been a topic of continuing investigation during the last years. In particular, significant progress has been made by the development of adaptive algorithms, which aim at the control of time and space grids in such a way that on one hand the solution is as accurate as required by the user and on the other hand the necessary work to obtain such a solution is minimized. The present paper surveys some of the essential features of such adaptive methods, which have been developed recently by the authors.

Parallel Extrapolation Methods and their Application in Chemical Engineering

We study the parallelization of linearly-implicit extrapolation methods for the solution of large scale systems of differential algebraic equations arising in a method of lines (MOL) treatment of partial differential equations. In our approach we combine a slightly overlapping domain decomposition together with a polynomial block Neumann preconditioner. Through the explicit computation of the matrix products of the preconditioner and the system matrix a significant gain in overall efficiency is achieved for medium-sized problems. The parallel algorithm exhibits a good scalability up to 32 processors on a Cray T3E. Preliminary results for computations on a workstation cluster are reported.

Eine graphische Oberfläche für numerische Programme

Bei der Weiterentwicklung von numerischen Verfahren im Scientific Computing werden haufig bereits existierende numerische Basis-Module modifiziert, sei es durch Verallgemeinerung oder Spezialisierung. Ein derartiges Vorgehen setzt gute Kenntnisse uber vorhandene (und zugangliche) Basis-Module und deren Arbeitsweise voraus.

Prognoserechnungen zur AIDS-Ausbreitung in der Bundesrepublik Deutschland

Die Ausbreitung der Immunschwachekrankheit AIDS ist fur unsere Gesellschaft ein bedrohliches Problem. Offentlichkeit, Politik und Wirtschaft drangen auf Prognoserechnungen. Wegen der zahlreichen „weichen“ Fakten in diesem Bereich ist jedoch eine mathematische Modellierung nicht ganz unproblematisch. Andererseits besteht langst nicht mehr die Freiheit zu wahlen, ob uberhaupt mathematische Modelle aufgestellt werden, sondern nur noch ob seriose mathematische Modelle aufgestellt werden. Bei Verzicht auf eine seriose Modellierung bleibt das Feld anderen uberlassen, die Vorhersagen ohne methodischen Unterbau abgeben — dazu ist aber die Fragestellung einfach zu wichtig.

An overview of MEXX: Numerical Software for the Integration of Multibody Systems

MEXX (short for MEXanical systems eXtrapolation integrator) is a Fortran code for time integration of constrained mechanical systems, which was developed at the University of Innsbruck and the Konrad-Zuse-Center Berlin. It is suited for direct integration of the equations of motion in descriptor form, and has the following features: n n nTime stepping by a half-explicit extrapolation method, allowing for the accurate and robust computation of position, velocity, acceleration, and constraint forces. n n nOnly position and velocity constraint functions are evaluated, acceleration constraints need not be formulated. n n nBoth position and velocity constraints remain satisfied throughout the integration interval. n n nUses well-structured linear algebra, enabling the use of O(n) recursive elimination, among other full and sparse linear algebra options. n n nTime-continuous solution representation (e.g., for graphics) n n nRoot-finding options (e.g., for impact and Coulomb friction problems) n n n nMEXX encourages the use of large, sparse descriptor formulations, but can also efficiently handle near-statespace kinematic formulations of multibody systems. A detailed description of MEXX and the underlying concepts is given in [5]. In the present short note we provide a brief overview.