TY - GEN A1 - Nowak, Ulrich A1 - Weimann, Lutz T1 - GIANT - A Software Package for the Numerical Solution of Very Large Systems of Highly Nonlinear Equations. N2 - This report presents the final realization and implementation of a global inexact Newton method proposed by Deuflhard. In order to create a complete piece of software, a recently developed iterative solver (program GBIT) due to Deuflhard, Freund, Walter is adapted and serves as the standard iterative linear solver. Alternative linear iterative solvers may be adapted as well, e.g. the widely distributed code GMRES. The new software package GIANT (Global Inexact Affine Invariant Newton Techniques) allows an efficient and robust numerical solution of very large scale highly nonlinear systems. Due to the user friendly interface and its modular design, the software package is open for an easy adaptation to specific problems. Numerical experiments for some selected problems illustrate performance and usage of the package. T3 - ZIB-Report - TR-90-11 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4746 ER - TY - GEN A1 - Caprasse, H. A1 - Demaret, J. A1 - Gatermann, Karin A1 - Melenk, Herbert T1 - Power-Law Type Solutions of Fourth-Order Gravity N2 - We study the power-law type solutions of the fourth order field equations derived from a generic quadratic Lagrangian density in the case of multidimensional Bianchi I cosmological models. All the solutions of the system of algebraic equations have been found, using computer algebra, from a search of the Groebner bases associated to it. While, in space dimension $ d = 3 $ , the Einsteinian Kasner metric is still the most general power-law type solution, for $ d > 3 $ , no solution, other than the Minkowski space-time, is common to the three systems of equations associated with the three contributions to the Lagrangian density. In the case of a pure Riemann-squared contribution (suggested by a recent calculation of the effective action for the heterotic string), the possibility exists to realize a splitting of the $ d $-dimensional space into a ( $ d - 3 $)-dimensional internal space and a physical 3- dimensional space, the latter expanding in time as a power bigger than 2 (about 4.5 when $ d = 9 $). T3 - ZIB-Report - SC-90-18 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-466 ER - TY - GEN A1 - Wang, Daoliu T1 - Symplectic Difference Schemes for Perturbed Hamiltonian Systems. N2 - In this paper we consider symplectic difference schemes for perturbed Hamiltonian systems of integrable ones, which can cover many important problems. Symplectic difference schemes for general Hamiltonian systems can also be used to these problems. But the perturbation property has not been paid proper attention to, which is important in the method proposed here. Numerical simulation shows that, for this method the time step size can be taken quite large and the qualitative property , such as preserving invariant tori, is also better than usual symplectic difference schemes. T3 - ZIB-Report - SC-90-17 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-456 ER - TY - GEN A1 - Hege, Hans-Christian A1 - Stüben, Hinnerk T1 - Vectorization and Parallelization of Irregular Problems via Graph coloring. N2 - Efficient implementations of irregular problems on vector and parallel architectures are generally hard to realize. An important class of problems are Gauß-Seidel iteration schemes applied to irregular data sets. The unstructured data dependences arising there prevent restructuring compilers from generating efficient code for vector or parallel machines. It is shown, how to structure the data dependences by decomposing the underlying data set using graph coloring techniques and by specifying a particular execution order already on the algorithm level. Methods to master the irregularities originating from different types of tasks are proposed. An application is given and some open issues and future developments are discussed. T3 - ZIB-Report - SC-90-16 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-443 ER - TY - GEN A1 - Roitzsch, Rainer A1 - Kornhuber, Ralf T1 - BOXES - a Program to Generate Triangulations from a Rectangular Domain Description. N2 - BOXES computes a triangulation from a 2D domain description which consists of an arbitrary set of rectangles. Each rectangle may have attributes to control the triangulating process, define subdomain classes, or specify boundary conditions. The output of the program can be used as a coarse grid for KASKADE or one of its variants. Additional features are extensive checking of the user input, graphical display, and simple editing. T3 - ZIB-Report - TR-90-09 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4738 ER - TY - GEN A1 - Ackermann, Jörg A1 - Wulkow, Michael T1 - MACRON - A Program Package for Macromalecular Reaction Kinetics. N2 - This paper presents the new program package MACRON for the simulation of macromolecular kinetics including standard chemical reactions. Such problems lead to countable (possibly) infinite systems of ordinary differential equations (CODE's), which are numerically treated by the so-called discrete Galerkin method here. By a chemical compiler the required analytical preprocessing is performed, such that the complete reaction system, standard kinetics as well as macromolecular reactions, can be entered in the chemical formalism. Typical macromolecular reaction steps are chain addition, termination, chain transfer and degradation (cracking). In order to ensure efficiency and reliability, high sophisticated numerical routines are built within the package. MACRON can be used without a detailed knowledge of the used numerical methods. As an illustration the application of MACRON to some realistic problems is presented. T3 - ZIB-Report - SC-90-14 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-423 ER - TY - GEN A1 - Wulkow, Michael A1 - Ackermann, Jörg T1 - Numerical Treatment of Polyreactions - Recent Developments. N2 - 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. T3 - ZIB-Report - SC-90-15 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-439 ER - TY - GEN A1 - Wulkow, Michael T1 - Numerical Treatment of Countable Systems of Ordinary Differential Equations. N2 - 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. T3 - ZIB-Report - TR-90-08 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4728 ER - TY - GEN A1 - Bornemann, Folkmar A. T1 - An Adaptive Multilevel Approach to Parabolic Equations II. N2 - In continuation of part I this paper develops a variable-order time discretization in Hilbert space based on a multiplicative error correction. Matching of time and space errors as explained in part I allows to construct an adaptive multilevel discretization of the parabolic problem. In contrast to the extrapolation method in time, which has been used in part I, the new time discretization allows to separate space and time errors and further to solve fewer elliptic subproblems with less effort, which is essential in view of the application to space dimension greater than one. Numerical examples for space dimension one are included which clearly indicate the improvement. T3 - ZIB-Report - SC-90-13 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-410 ER - TY - GEN A1 - Walter, Artur T1 - Improvement of Incomplete Factorizations by a Sparse Secant Method. N2 - In the present paper, the improvement of an incomplete factorization of a non-symmetric matrix A is discussed. Starting from the ideas of sparsity preserving quasi-Newton methods, an algorithm is developed which improves the approximation of A by the incomplete factorization maintaining the sparsity structure of the matrices. No renumbering of the unknowns or the admittance of additional fill-in is necessary. The linear convergence of the algorithm is proved under the assumption, that $ L $ and $ U $* have the same sparsity structure and an incomplete factorization with some reasonable approximation property exits. In combination with this algorithm, the method of incomplete factorization and its several modifications are applicable to a wider class of problems with improved convergence qualities. This is shown by a numerical example. {\bf Key Words:} non-symmetric linear system, sparse secant method, incomplete factorization. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20, 65N30. T3 - ZIB-Report - SC-90-12 KW - non-symmetric linear systems KW - sparse secant method KW - incomplete factorization Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-401 ER -