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Der folgende Bericht ist eine Ergänzung des ZIB Technical Report TR 88-05. Entsprechend wird hier nicht auf die grundsätzlichen Fragen der Parallelisierbarkeit des sequentiellen TRAPEX eingegangen. Diese sind im TR 88-5 erörtert, die dort beschriebenen Algorithmen (vertikale und horizontale Parallelisierung) werden auch für die Transputerarchitektur verwendet. Meßergebnisse sind im letzten Teil angefügt.
Die Parallelisierung eines vorhandenen sequentiellen Programmes erfolgt im allgemeinen in der Weise, daß es auf Grund einer Analyse der Datenabhängigkeiten in mehrere parallel ausführbare Teile zerlegt wird, die ihrerseits sequentiell ablaufen und untereinander Daten austauschen. Eine parallele Programmiersprache sollte diese Vorgehensweise unterstützen, indem sie Sprachmittel zur Verfügung stellt, mittels derer sich die Modularisierung und Kommunikation bequem formulieren läßt. Nun läßt sich prinzipiell jede vorhandene sequentielle Programmiersprache um solche Konstrukte erweitern; günstiger in Hinblick auf die Strukturierung eines parallelen Programmes erscheint es aber möglicherweise, eine solche Programmiersprache zu verwenden, die die genannten Konzepte als elementare zur Verfügung stellt. Dies ist offensichtlich bei objektorientierten Programmiersprachen der Fall. Im vorliegenden Bericht wird an Hand eines Beispiels - TRAPEX -, das in POOL-T implementiert wurde, untersucht, inwieweit eine objektorientierte Sprache zur effektiven Programmierung eines numerischen Algorithmus geeignet ist. {\bf Key Words:} parallele objektorientierte Programmiersprache, Message Passing, Abstract Data Typing, Klassenhierarchi, Modul-Import-Mechanismus, Client/Server-Prinzip, Load Balancing, adaptive numerische Algorithmen, Ordnungs- und Schrittweitensteuerung, Romberg-Quadratur.
Der Bericht ist die Ausarbeitung eines Vortrages, der auf dem Transputer Anwender Treffen (TAT'89) im September 1989 gehalten wurde. Es wird die Parallelisierung und Implementierung eines adaptiven Algorithmus zur numerischen Integration (Romberg Quadratur) beschrieben. Ausgewählte Meßergebnisse sind enthalten. {\bf Keywords:} Numerischer Algorithmus, Romberg Quadratur, paralleler adaptiver Algorithmus, dynamische Lastverteilung und Prozessorzahl, lokaler Speicher, Nearest-Neighbour-Architektur, Transputer, TDS, OCCAM2.
Adaptive multilevel discretization in time and space for parabolic partial differential equations.
(1989)
The present paper developes an adaptive multilevel approach for parabolic PDE's - as a first step, for one linear scalar equation. Full adaptivity of the algorithm is conceptually realized by simultaneous multilevel discretization in both time and space. Thus the approach combines multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization such as the hierarchical basis method. The algorithmic approach is theoretically backed by careful application of fundamental results from semigroup theory. These results help to establish the existence of asymptotic expansions (in terms of time-steps) in Hilbert space. Finite element approximation then leads to perturbed expansions, whose perturbations, however, can be pushed below a necessary level by means of an adaptive grid control. The arising space grids are not required to satisfy any quasi- uniformity assumption. Even though the theoretical presentation is independent of space dimension details of the algorithm and numerical examples are given for the 1-D case only. For the 1-D elliptic solver, which is used, an error estimator is established, which works uniformly well for a family of elliptic problems. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.
In this paper we present the MmB schemes, which preserve the local maximum and minimum bounds of the initial data in the smallest union of mesh elements of previous time step containing the domain of dependence of the solution on the mesh element with center at point $ P $\ under consideration. In 1-D, the MmB schemes are almost identical with TVD schemes. As well-known, there is no second-order TVD scheme in 2-D, nevertheless, we present here two classes of 2-D second-order accurate MmB-schemes. It is proved that 1-D discrete MmB (or TVD) and 1-D semi-discrete TVD schemes may have second-order accuracy at (nonsonic) critical points, but cannot be of uniformly second-order accurate in the whole neighborhood of the critical points. New accurate high resolution flux limiters are suggested. Numerical results for 1-D and 2-D test problems are given. {\bf Keywords:} Difference scheme, TVD, MmB, flux limiter.
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.
The SPARC processor is a RISC (Reduced Instruction Set Computer) microcomputer, built into the SUN4 workstations. Since RISC processors are very well-suited for LISP processing, the implementation of a dialect of LISP (Portable Standard LISP, PSL) boded well for a great speed-up in comparison with other types of microcomputers. A first approach was done at The RAND Corporation in Santa Monica, which was derived from classical processor types like MC68000 or VAX. At the Konrad- Zuse-Zentrum für Informationstechnik Berlin (ZIB) that initial implementation was redesigned in order to adapt PSL to the specific features of the SPARC processor. The present implementation, in some parts, is very close to Cray PSL version also done in ZIB. Some timing informations are given in the appendix.
The paper presents a detailed analysis of the possible accuracy available for TVD schemes in one dimension with emphasis to the semi-discrete 1-D TVD schemes. The analysis shows that the widely accepted statement [1] of degeneration of accuracy at critical points for TVD schemes should be corrected. We have theorem: TVD schemes using flux limiters $ \varphi $ of the form [1], [2] may be second-order accurate at critical points if $ \varphi $ (3) + $ \varphi $(-1) = 2, but cannot be uniformly second-order accurate in the whole neighborhood of critical point. If $ \varphi $(1) = 1, then the TVD schemes are second-order accurate in the region of smooth solutions sufficiently far from the critical points. Two ways are suggested to improve the accuracy. Numerical example is given. {\bf Keywords:} Semi-discrete schemes, TVD, flux limiter, degeneration of accuracy.
In this paper we consider the problem of reconstructing a multivariate rational function, when only its values at sufficiently many points are known. We use for the reconstruction of bivariate rational functions a bivariate rational interpolation operator investigated by Siemaszko [7] and a new one, compare both by examples in a Computer Algebra system, and present their multivariate generalizations. {\bf Keywords:} Multivariate rational interpolation, reconstruction, symbolic computation.
We propose an anisotropic refinement strategy which is specially designed for the efficient numerical resolution of internal and boundary layers. This strategy is based on the directed refinement of single triangles together with adaptive multilevel grid orientation. It is demonstrated by several numerical examples that compared to usual methods, the new anisotropic refinement ends up in more stable and more accurate solutions at much less computational cost. {\bf Keywords:} Adaptive finite elements, directed refinement, adaptive grid orientation, convection diffusion equation, internal and boundary layers.