@misc{Wulkow1991, author = {Wulkow, Michael}, title = {Adaptive Treatment of Polyreactions in Weighted Sequence Spaces.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-673}, number = {SC-91-17}, year = {1991}, abstract = {Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and statistics. They can be considered as ordinary differential equations in sequence spaces. In this work, a fully adaptive algorithm for the computational treatment of such systems is developed. The method is based on a time discretization of an abstract Cauchy problem in Hilbert space and a discrete Galerkin approach for the discretization of the arising stationary subproblems. The Galerkin method uses orthogonal functions of a discrete variable, which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations are solved adaptively either by use of analytical properties of the orthogonal functions or by an appropriate numerical summation. The resulting algorithm CODEX is applied to examples of technological interest, in particular from polymer chemistry.}, language = {en} } @misc{AckermannWulkow1991, author = {Ackermann, J{\"o}rg and Wulkow, Michael}, title = {The Treatment of Macromolecular Processes with Chain-Length-Dependent Reaction Coefficients - An Examplefrom Soot Formation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-682}, number = {SC-91-18}, year = {1991}, abstract = {The description of chain length distributions in macromolecular reaction kinetics leads to so-called countable systems of differential equations. In particular, when the appearing reaction rate coefficients depend on the chain length of the reacting macromolecules itself, an efficient numerical treatment of these systems is very difficult. Then even the evaluation of the right-hand side of the system can become prohibitively expensive with respect to computing time. In this paper we show how the discrete Galerkin method can be applied to such problems. The existing algorithm CODEX is improved by use of a multiplicative error correction scheme for time discretization and a new type of numerical preprocessing by means of a Gauss summation. Both ideas are exemplary for a wide class of approximation types and are described very briefly here. The new numerical techniques are tested on an example from soot formation, where the coagulation of molecules is modeled in terms of reaction coefficients depending on the surface of the particles and their collision frequency.}, language = {en} } @misc{DeuflhardWulkow1988, author = {Deuflhard, Peter and Wulkow, Michael}, title = {Computational Treatment of Polyreaction Kinetics by Orthogonal Polynomials of a Discrete Variable.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-140}, number = {SC-88-06}, year = {1988}, abstract = {The paper presents a new approach to the computational treatment of polyreaction kinetics. This approach is characterized by a Galerkin method based on orthogonal polynomials of a discrete variable, the polymer degree (or chain length). In comparison with the known competing approaches (statistical moment treatment, Galerkin methods for continuous polymer models), the suggested method is shown to avoid the disadvantages and preserve the adventages of either of them. The basic idea of the method is the construction of a discrete inner product associated with a reasonably chosen probability density function. For the so-called Schulz-Flory distribution one thus obtains the discrete Laguerre polynomials, whereas the Poisson distribution leads to the Charlier polynomials. Numerical experiments for selected polyreaction mechanisms illustrate the efficiency of the proposed method.}, language = {en} } @misc{SchuetteWulkow1992, author = {Sch{\"u}tte, Christof and Wulkow, Michael}, title = {Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of RamanSpectroscopy.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-778}, number = {SC-92-07}, year = {1992}, abstract = {Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown.}, language = {en} } @misc{BuddeWulkow1989, author = {Budde, Uwe and Wulkow, Michael}, title = {Computation of Molecular Weight Distributions for Free Radical Polymerization Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-253}, number = {SC-89-07}, year = {1989}, abstract = {Modeling of free radical polymerization leads to very large and usually stiff systems of ordinary differential equations which cannot be solved directly in an efficient way. This paper presents the application of a new approach called discrete Galerkin method to a realistic example - the polymerization of methyl methacrylate(MMA). The method is characterized by a Galerkin approximation on the basis of orthogonal polynomials of a discrete variable which represents the polymer degree. It allows the efficient computation of solutions of complete kinetic schemes with time- or moment-dependent reaction coefficients by reducing the complexity to a few differential equations. The approximation error can be controlled by an error estimation. In the case of MMA polymerization a reduction of computational effort by a factor of about 25 compared to a standard method can be obtained for the quasi-steady-state approximation of the model. In addition solutions of the instationary kinetic scheme can be easily computed.}, language = {en} } @misc{WulkowDeuflhard1990, author = {Wulkow, Michael and Deuflhard, Peter}, title = {Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-292}, number = {SC-90-01}, year = {1990}, abstract = {The discrete Galerkin method developed by the authors has turned out to be an efficient tool for the computational treatment of very large scale ODE systems arising in polyreaction kinetics. Up to now, this approach has been worked out in detail for homogeneous polymer reactions. The present paper deals with one line of possible extensions of the method to the case of so-called heterogeneous processes, which may appear e. g. in smog reactions. The associated mathematical models involve reaction coefficients depending on the chain length of the reacting polymer. The herein suggested extension is worked out in some detail on the basis of the earlier paper. In addition, a numerical example describing polymer degradation is included.}, language = {en} } @misc{DeuflhardNowakWulkow1990, author = {Deuflhard, Peter and Nowak, Ulrich and Wulkow, Michael}, title = {Recent Developments in Chemical Computing.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-359}, number = {SC-90-07}, year = {1990}, abstract = {The 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 methods 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.}, language = {en} } @misc{Wulkow1990, author = {Wulkow, Michael}, title = {Numerical Treatment of Countable Systems of Ordinary Differential Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4728}, number = {TR-90-08}, year = {1990}, abstract = {Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and medicine. They can be considered as ordinary differential equations in sequence spaces. In this work, a full adaptive algorithm for the computational treatment of such systems is developed. The method combines time discretization with extrapolation in Hilbert spaces with a discrete Galerkin approach as discretization of the stationary subproblems. The Galerkin method is based on orthogonal functions of a discrete variable , which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations can be assembled either by use of analytical properties of the orthogonal functions or numerically by a multilevel summation algorithm. The resulting algorithm CODEX is applied to many examples of technological interest, in particular from polymer chemistry.}, language = {en} } @article{DeuflhardNowakWulkow1990, author = {Deuflhard, Peter and Nowak, Ulrich and Wulkow, Michael}, title = {Recent Developments in Chemical Computing}, volume = {14}, journal = {Computers in Chemical Engineering}, pages = {1249 -- 1258}, year = {1990}, language = {en} }