@misc{DeuflhardLangNowak1995, author = {Deuflhard, Peter and Lang, Jens and Nowak, Ulrich}, title = {Adaptive Algorithms in Dynamical Process Simulation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1823}, number = {SC-95-16}, year = {1995}, abstract = {Dynamical simulation of industrially relevant processes strongly advises the use of algorithms, which are {\em adaptive} both in time and in space discretization. The paper presents two alternatives: (a) a fully adaptive method of lines approach, which is based on finite difference methods and essentially applicable to 1D problems; (b) a fully adaptive Rothe method, which is based on a fast multilevel finite element method and applicable to 1D up to 3D.}, language = {en} } @misc{FroehlichDeuflhard1994, author = {Fr{\"o}hlich, Jochen and Deuflhard, Peter}, title = {Moving Weight Galerkin Methods for Turbulent Reactive Flows}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1646}, number = {SC-94-36}, year = {1994}, abstract = {Adopting a statistical approach for the computation of turbulent combustion flows an approximation for the probability density function (PDF) of the composition variables is often required to treat the highly non-linear reaction term in a satisfactory way. One class of methods currently being used are the moment methods which employ transport equations for low order statistical moments and use a parametrized shape of the PDF. A second class solves a transport equation for the joint PDF by a Monte Carlo method. In the present paper we develop an intermediate algorithm based on a Galerkin method for the PDF transport equation. The solution is developed in terms of an orthogonal or bi-orthogonal basis of a suitable Hilbert space. The unconventional use of the related weight function as a prefactor (moving weight approach) permits adaptivity and results in a generalization of the \$\beta-\$closure for bounded scalar quantities. We present the approximation procedure in detail and apply it to the evolution of the composition in a homogeneous well-stirred reactor. The extension to non-homogeneous flow simulations is straightforward.}, language = {en} } @misc{DeuflhardAckermann1993, author = {Deuflhard, Peter and Ackermann, J{\"o}rg}, title = {Adaptive Discrete Galerkin-Methods for Macromolecular Processes.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1180}, number = {SC-93-22}, year = {1993}, abstract = {In this paper, a rather recent algorithmic approac to the numerical simulation of macromolecula processes is surveyed. It avoids the numerical stiff integration o thousands up to millions of ODE's by constructing a scale of discret Hilbert spaces, especially weighted sequence spaces, and establishing corresponding Galerkin method. Examples including polyreactions o industrial relevance and ecological waste management by biochemica recycling illustrate the importance and efficiency of the algorithm.}, language = {en} } @misc{Deuflhard1993, author = {Deuflhard, Peter}, title = {Cascadic Conjugate Gradient Methods for Elliptic Partial Differential Equations I. Algorithm and Numerical Results.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1197}, number = {SC-93-23}, year = {1993}, abstract = {Cascadic conjugate gradient methods for the numerical solution of elliptic partial differential equations consists of Galerkin finite element methods as outer iteration and (possibly preconditioned) conjugate gradient methods as inner iteration. Both iterations are known to minimize the energy norm of the arising iterations errors. A simple but efficient strategy to control the discretization errors versus the PCG iteration errors in terms of energy error norms is derived and worked out in algorithmic detail. In a unified setting, the relative merits of different preconditioners versus the case of no preconditioning is compared. Surprisingly, it appears that the cascadic conjugate gradient method without any preconditioning is not only simplest but also fastest. The numerical results seem to indicate that the cascade principle in itself already realizes some kind of preconditioning. A theoretical explanation of these observations will be given in Part II of this paper.}, language = {en} } @misc{DeuflhardSchmidt1993, author = {Deuflhard, Peter and Schmidt, Frank}, title = {Discrete Transparent Boundary Conditions for the Numerical Solution of Fresnel's Equation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1205}, number = {SC-93-24}, year = {1993}, abstract = {The paper presents a construction scheme of deriving transparent , i. e. reflection-free, boundary conditions for the numerical solution of Fresnel's equation (being formally equivalent to Schr{\"o}dinger's equation). These boundary conditions appear to be of a nonlocal Cauchy type. As it turns out, each kind of linear implicit discretization induces its own discrete transparent boundary conditions.}, language = {en} } @misc{WulffHohmannDeuflhard1994, author = {Wulff, Claudia and Hohmann, Andreas and Deuflhard, Peter}, title = {Numerical Continuation of Periodic Orbits with Symmetry.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1421}, number = {SC-94-12}, year = {1994}, abstract = {We consider periodic orbits of autonomous parameter dependent ODE's. Starting from a shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincar\'e-section we develop a pathfollowing algorithm for periodic solutions based on a tangential continuation method with implicit reparametrization. For ODE's equivariant w.r.t. a finite group we show that spatial as well as spatio-temporal symmetries of periodic orbits can be exploited within the (multiple) shooting context. We describe how turning points, period doubling bifurcations and Hopf points along the branch of periodic solutions can be handled. Furthermore equivariant Hopf points and generic secondary bifurcations of periodic orbits with \$ Z_m\$-symmetry are treated. We tested the code with standard examples, e.g., the period doubling cascade in the Lorenz equations. To show the efficiency of the described methods we also used the program for an application from electronics, a ring oscillator with \$n \$ inverters. In this example the exploitation of symmetry reduces the amount of work for the continuation of periodic orbits from \${\cal O}(n^2)\$ to \${\cal O}(n)\$}, language = {en} } @misc{DeuflhardWulkow1994, author = {Deuflhard, Peter and Wulkow, Michael}, title = {Simulationsverfahren f{\"u}r die Polymerchemie}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1502}, number = {SC-94-22}, year = {1994}, abstract = {Die enorme algorithmische Beschleunigung durch diskrete Galerkin-Methoden f{{\"u}}r abz{{\"a}}hlbare Differentialgleichungssysteme hat der Simulation von Polymerisationsprozessen neue, industriell relevante Problembereiche er{{\"o}}ffnet, die mit den bis dahin verf{{\"u}}gbaren Methoden nicht zug{{\"a}}nglich waren.}, language = {de} } @misc{DeuflhardNowakWeyer1994, author = {Deuflhard, Peter and Nowak, Ulrich and Weyer, J.}, title = {Prognoserechnung zur AIDS-Ausbreitung in der Bundesrepublik Deutschland}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1511}, number = {SC-94-23}, year = {1994}, abstract = {Die Autoren sind sich dar{{\"u}}ber im klaren, daß zahlreiche Imponderabilien die Ergebnisse der vorgelegten Modellrechnungen beeinflussen k{{\"o}}nnen. Immerhin repr{\"a}sentiert jedoch das vorgelegte mathematische Modell die Summe unserer derzeitigen Kenntnisse zur Fragestellung AIDS--Epidemie; neu hinzukommende Erkenntnisse lassen sich rasch einarbeiten und in ihren Konsequenzen {\"u}berschauen. Die Entwicklung neuer effizienter numerischer Methoden spielte eine Schl{\"u}sselrolle bei der tats{\"a}chlichen Simulation dieses umfangreichen realit{\"a}tsnahen mathematischen Modells.}, language = {de} } @misc{Deuflhard1993, author = {Deuflhard, Peter}, title = {A Study of Lanczos-Type Iterations for Symmetric Indefinite Linear Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1029}, number = {SC-93-06}, year = {1993}, abstract = {The Lanczos iteration for symmetric indefinite linear systems seems to be well--known for quite a while. However, in order to modify it with the aim of improved performance, the present paper studies certain aspects in terms of an adjoint scalar three--term recurrence. Thus, at least a different view is opened. Moreover, an alternative \$3n\$--implementation in terms of the Euclidean orthogonal basis has been found that easily permits generalizations. The study is understood as a start--off for further numerical investigations and experiments.}, language = {en} } @misc{SchoepfDeuflhard1991, author = {Sch{\"o}pf, R. and Deuflhard, Peter}, title = {OCCAL: A mixed symbolic- numeric Optimal Control CALculator.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-638}, number = {SC-91-13}, year = {1991}, abstract = {The numerical solution of optimal control problems by indirect methods (such as multiple shooting or collocation) requires a considerable amount of analytic calculation to establish a numerically tractable system. These analytic calculations, though being rather tedious in realistic examples, are nowadays mostly still done by hand-and thus prone to calculation errors. The paper aims at automating this analytic processing to a reasonable extent by means of a modern symbolic manipulation language (here: REDUCE). In its present stage of development the package OCCAL (mnemotechnically for \underline{O}ptimal \underline{C}ontrol \underline{CAL}culator) permits an interactive use, covering tasks like automatic determination of control and, in case of a singular control, of its order. In simpler problems, the present version of OCCAL automatically produces the full subroutine input for a MULtiple shooting code (MULCON) with adaptive numerical CONtinuation. In more complicate problems where singular sub-arcs may occur or where the sequence of sub-arcs of the optimal trajectory is unclear OCCAL is a significant help in reducing analytic pre-processing. Numerical examples illustrate the performance of OCCAL/MULCON.}, language = {en} } @misc{DeuflhardLeinenYserentant1988, author = {Deuflhard, Peter and Leinen, P. and Yserentant, Harry}, title = {Concepts of an Adaptive Hierarchical Finite Element Code.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-131}, number = {SC-88-05}, year = {1988}, abstract = {The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device - which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved - independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local - making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem. {\bf Keywords:} finite elements, hierarchical basis, adaptive mesh refinement, preconditioned conjugate gradient methods.}, language = {en} }