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  • 1990-1994  (12)
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  • 1
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    The Treatment of Macromolecular Processes with Chain-Length-Dependent Reaction Coefficients - An Examplefrom Soot Formation. (1991)
    Details
    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 2
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    Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of RamanSpectroscopy. (1992)
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    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 3
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    Numerical Treatment of Countable Systems of Ordinary Differential Equations. (1990)
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    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 4
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    Recent Developments in Chemical Computing (1990)
    Details
    Publication Date: 2020-12-14
    Language: English
    Type: article , doc-type:article
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  • 5
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    Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions (1992)
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    Publication Date: 2020-03-20
    Language: English
    Type: bookpart , doc-type:bookPart
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  • 6
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    Recent Developments in Chemical Computing. (1990)
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    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 7
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    Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions. (1990)
    Details
    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 8
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    Numerical Treatment of Polyreactions - Recent Developments. (1990)
    Details
    Publication Date: 2014-02-26
    Description: The mathematical modeling of macromolecular reactions leads to countable (possibly infinite) systems of ordinary differential equations (CODE's). This paper reviews two recent developments of the so-called discrete Galerkin method, which has been developed for the numerical treatment of countable systems, which arise e.g. in polymer chemistry. The first approach can be considered as a method of lines with moving basis functions and has been implemented recently in the program package MACRON. The second type of the Galerkin method is characterized by a so-called outer time discretization of the complete problem and an appropriate and efficient solution of the arising subproblems. This method is realized in the research code CODEX.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 9
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    Adaptive Treatment of Polyreactions in Weighted Sequence Spaces. (1991)
    Details
    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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  • 10
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    Simulationsverfahren für die Polymerchemie (1994)
    Details
    Publication Date: 2014-02-26
    Description: Die enorme algorithmische Beschleunigung durch diskrete Galerkin-Methoden f{ü}r abz{ä}hlbare Differentialgleichungssysteme hat der Simulation von Polymerisationsprozessen neue, industriell relevante Problembereiche er{ö}ffnet, die mit den bis dahin verf{ü}gbaren Methoden nicht zug{ä}nglich waren.
    Keywords: ddc:000
    Language: German
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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