TY - GEN A1 - Maas, Ulrich A1 - Nowak, Ulrich T1 - An adaptive method of lines for the simulation of complex laminar combustion processes N2 - For the simulation of one-dimensional flame configurations reliabl e numerical tools are needed which have to be both highly efficient (large num ber of parametric calculations) and at the same time accurate (in order t o avoid numerical errors). This can only be accomplished using fully adapt ive discretization techniques both in space and time together with a c ontrol of the discretization error. We present a method which accomplishes this task. It is based on a n adative MOL (method of lines) treatment. Space discretization is done by means of finite difference approxi mations on non-uniform grids. Time is discretized by the linearly-implicit Euler method. In order to control the discretization errors an extrapolation pro cedure is used in space and time. Results are presented for simple laser-induced ignition processes. The method, however, can be applied to other combustion processes, too. T3 - ZIB-Report - SC-96-56 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2665 ER - TY - GEN A1 - Ehrig, Rainald A1 - Nowak, Ulrich T1 - Algorithmen- und Softwareoptimierung für die Satellitenbahn- und Schwerefeldmodellierung N2 - Dieser Report enthält die Ergebnisse der Untersuchungen, die gemäss dem Forschungs-- und Entwicklungsvertrag Gravity zwischen dem GeoForschungsZentrum Potsdam und dem Konrad--Zuse--Zentrum Berlin vorgenommen wurden. Die damit vereinbarte wissenschaftliche Kooperation hat die folgenden Ziele: \item{die am GFZ vorhandenen Algorithmen und Softwaremodule auf ihre Effizienz hisichtlich Nutzung der Rechnerresourcen zu untersuchen und Lösungen für einen schnelleren Datendurchsatz zu entwickeln und zu implementieren,} \item{Methoden zur Regularisierung und Lösung schlecht konditionierter Normalgleichungssysteme (für Schwerefeldkoeffizienten) kritisch zu untersuchen und eine mathematisch objektive Strategie der Regularisierung zu entwickeln, und} \item{insbesondere im Hinblick auf die Anforderungen bei GRACE, verschiedene Bahnintegrationsverfahren hinsichtlich ihrer numerischen Genauigkeit und Einsatzmöglichkeiten zu untersuchen.} T3 - ZIB-Report - 00-01 KW - geodesy KW - gravity field models KW - regularization KW - ill-posed systems KW - orbit integration Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5694 ER - TY - GEN A1 - Schlegel, Martin A1 - Marquardt, Wolfgang A1 - Ehrig, Rainald A1 - Nowak, Ulrich T1 - Sensitivity analysis of linearly-implicit differential-algebraic systems by one-step extrapolation N2 - In this work we present an approach for the sensitivity analysis of linearly-implicit differential-algebraic equation systems. Solutions for both, states and sensitivities are obtained by applying an extrapolated linearly implicit Euler discretization scheme. This approach is compared to the widely used sensitivity extensions of multi-step BDF methods by means of case studies. Especially, we point out the benefit of this method in the context of dynamic optimization using the sequential approach. T3 - ZIB-Report - 02-38 KW - sensitivity evaluation KW - differential-algebraic systems KW - extrapolation methods KW - dynamic optimization KW - optimal control Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7052 ER - TY - GEN A1 - Deuflhard, Peter A1 - Nowak, Ulrich A1 - Weiser, Martin T1 - Affine Invariant Adaptive Newton Codes for Discretized PDEs N2 - The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and inexact Newton methods, where an inner iteration is used instead, are synoptically presented, both in affine invariant convergence theory and in numerical experiments. The three types of algorithms are: (a) affine covariant (formerly just called affine invariant) Newton algorithms, oriented toward the iterative errors, (b) affine contravariant Newton algorithms, based on iterative residual norms, and (c) affine conjugate Newton algorithms for convex optimization problems and discrete nonlinear elliptic PDEs. T3 - ZIB-Report - 02-33 KW - Affine invariant Newton methods KW - global Newton methods KW - inexact Newton methods KW - adaptive trust region methods KW - nonlinear partial differential equa Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7005 ER - TY - GEN A1 - Nowak, Ulrich A1 - Grah, Aleksander A1 - Schreier, Manfred T1 - Parameter Estimation and Accuracy Matching Strategies for 2-D Reactor Models N2 - The mathematical modeling of a special modular catalytic reactor kit leads to a system of partial differential equation in two space dimensions. As customary, this model contains unconfident physical parameters, which may be adapted to fit experimental data. To solve this nonlinear least squares problem we apply a damped Gauss-Newton method. A method of lines approach is used to evaluate the associated model equations. By an a priori spatial discretization a large DAE system is derived and integrated with an adaptive, linearly-implicit extrapolation method. For sensitivity evaluation we apply an internal numerical differentiation technique, which reuses linear algebra information from the model integration. In order not to interfere the control of the Gauss-Newton iteration these computations are done usually very accurately and, therefore, very costly. To overcome this difficulty, we discuss several accuracy adaptation strategies, e.g., a master-slave mode. Finally, we present some numerical experiments. T3 - ZIB-Report - 03-52 KW - parameter estimation KW - accuracy matching KW - method of lines KW - differential-algebraic systems KW - extrapolation methods KW - catalytic reactor models Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7749 ER - TY - GEN A1 - Deuflhard, Peter A1 - Nowak, Ulrich A1 - Lutz-Westphal, Brigitte T1 - Bessel'scher Irrgarten N2 - Dieser Artikel berichtet über eine erfolgreiche Schüleraktivität, die seit Jahren am Zuse-Institut Berlin (ZIB) bei Besuchen von Schülergruppen erprobt und verfeinert worden ist. Das hier zusammengestellte Material ist gedacht als Basis für eine Unterrichtseinheit in Leistungskursen Mathematik an Gymnasien. Inhaltlich wird von einem zwar für Schüler (wie auch Lehrer) neuen, aber leicht fasslichen Gegenstand ausgegangen: der Drei-Term-Rekursion für Besselfunktionen. Die Struktur wird erklärt und in ein kleines Programm umgesetzt. Dazu teilen sich die Schüler selbstorganisierend in Gruppen ein, die mit unterschiedlichen Taschenrechnern "um die Wette" rechnen. Die Schüler und Schülerinnen erfahren unmittelbar die katastrophale Wirkung von an sich kleinen'' Rundungsfehlern, sie landen -- ebenso wie der Supercomputer des ZIB -- im Bessel'schen Irrgarten''. Die auftretenden Phänomene werden mathematisch elementar erklärt, wobei lediglich auf das Konzept der linearen Unabhängigkeit zurückgegriffen wird. Das dabei gewonnene vertiefte Verständnis fließt ein in die Konstruktion eines klassischen Algorithmus sowie eines wesentlich verbesserten Horner-artigen Algorithmus. T3 - ZIB-Report - 07-09 KW - Rundungsfehler KW - Drei-Term-Rekursionen KW - Bessel-Rekursion Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9525 ER - TY - GEN A1 - Nowak, Ulrich A1 - Pöhle, Uwe A1 - Roitzsch, Rainer A1 - Werk, Roland T1 - ZGUI-Handbuch N2 - In diesem Handbuch werden die Bausteine zum Aufbau einer graphischen Benutzeroberfläche mit {\tt ZGUI} beschrieben. Auf der einen Seite stehen die Tcl/Tk--Prozeduren, die die graphischen Elemente definieren. Die Beschreibung der Anwendung der Prozeduren und der Interaktionen der Elemente bildet den ersten Teil des Handbuches. Auf der anderen Seite stehen die Anforderungen an Anwendungen, die mit einer {\tt ZGUI}--Benutzeroberlfäche gesteuert werden sollen. Hier findet man im Handbuch die Beschreibung der Anwendungsprogrammierschnittstelle (application programming interface, API). T3 - ZIB-Report - TR-96-10 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5397 ER - TY - GEN A1 - Ehrig, Rainald A1 - Nowak, Ulrich A1 - Deuflhard, Peter T1 - Highly scalable parallel linearly-implicit extrapolation algorithms N2 - We present parallel formulations of the well established extrapolation algorithms EULSIM and LIMEX and its implementation on a distributed memory architecture. The discretization of partial differential equations by the method of lines yields large banded systems, which can be efficiently solved in parallel only by iterative methods. Polynomial preconditioning with a Neumann series expansion combined with an overlapping domain decomposition appears as a very efficient, robust and highly scalable preconditioner for different iterative solvers. A further advantage of this preconditioner is that all computation can be restricted to the overlap region as long as the subdomain problems are solved exactly. With this approach the iterative algorithms operate on very short vectors, the length of the vectors depends only on the number of gridpoints in the overlap region and the number of processors, but not on the size of the linear system. As the most reliable and fast iterative methods based on this preconditioning scheme appeared GMRES or FOM and BICGSTAB. To further reduce the number of iterations in GMRES or FOM we can reuse the Krylov-spaces constructed in preceeding extrapolation steps. The implementation of the method within the program LIMEX results in a highly parallel and scalable program for solving differential algebraic problems getting an almost linear speedup up to 64 processors even for medium size problems. Results are presented for a difficult application from chemical engineering simulating the formation of aerosols in industrial gas exhaust purification. T3 - ZIB-Report - TR-96-11 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5405 ER - TY - THES A1 - Nowak, Ulrich T1 - Adaptive Linienmethoden für nichtlineare parabolische Systeme in einer Raumdimension. N2 - A new method for the numerical solution of highly nonlinear, coupled systems of parabolic differential equations in one space dimension is presented. The approach is based on a classical method of lines treatment. Time discretization is done by means of the semi--implicit Euler discretization. Space discretization is done with finite differences on non--uniform grids. Both basic discretizations are coupled with extrapolation techniques. With respect to time the extrapolation is of variable order whereas just one extrapolation step is done in space. Based on local error estimates for both, the time and the space discretization error, the accuracy of the numerical approximation is controlled and the discretization stepsizes are adapted automatically and simultaneously. Besides the local adaptation of the space grids after each integration step (static regridding), the grid may even move within each integration step (dynamic regridding). Thus, the whole algorithm has a high degree of adaptivity. Due to this fact, challenging problems from applications can be solved in an efficient and robust way. T3 - ZIB-Report - TR-93-14 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5059 ER - 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 -