Adaptive Multiscale Schemes for Conservation Laws

Softcover Nachdruck of the original 1. Auflage 2003
ISBN: 978-3-540-44325-4
Verlag: Springer Berlin Heidelberg

Buch, Englisch, Band 27, 188 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 318 g

Reihe: Lecture Notes in Computational Science and Engineering

During the last decade enormous progress has been achieved in the field of computational fluid dynamics. This became possible by the development of robust and high-order accurate numerical algorithms as well as the construc­ tion of enhanced computer hardware, e. g., parallel and vector architectures, workstation clusters. All these improvements allow the numerical simulation of real world problems arising for instance in automotive and aviation indus­ try. Nowadays numerical simulations may be considered as an indispensable tool in the design of engineering devices complementing or avoiding expen­ sive experiments. In order to obtain qualitatively as well as quantitatively reliable results the complexity of the applications continuously increases due to the demand of resolving more details of the real world configuration as well as taking better physical models into account, e. g., turbulence, real gas or aeroelasticity. Although the speed and memory of computer hardware are currently doubled approximately every 18 months according to Moore's law, this will not be sufficient to cope with the increasing complexity required by uniform discretizations. The future task will be to optimize the utilization of the available re­ sources. Therefore new numerical algorithms have to be developed with a computational complexity that can be termed nearly optimal in the sense that storage and computational expense remain proportional to the "inher­ ent complexity" (a term that will be made clearer later) problem. This leads to adaptive concepts which correspond in a natural way to unstructured grids.

Zielgruppe

Research

Autoren/Hrsg.

Müller, Siegfried

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik

Inhaltsverzeichnis

1 Model Problem and Its Discretization.- 1.1 Conservation Laws.- 1.2 Finite Volume Methods.- 2 Multiscale Setting.- 2.1 Hierarchy of Meshes.- 2.2 Motivation.- 2.3 Box Wavelet.- 2.4 Change of Stable Completion.- 2.5 Box Wavelet with Higher Vanishing Moments.- 2.6 Multiscale Transformation.- 3 Locally Refined Spaces.- 3.1 Adaptive Grid and Significant Details.- 3.2 Grading.- 3.3 Local Multiscale Transformation.- 3.4 Grading Parameter.- 3.5 Locally Uniform Grids.- 3.6 Algorithms: Encoding, Thresholding, Grading, Decoding.- 3.7 Conservation Property.- 3.8 Application to Curvilinear Grids.- 4 Adaptive Finite Volume Scheme.- 4.1 Construction.- 4.2 A gorithms: Initial data, Prediction, Fluxes and Evolution.- 5 Error Analysis.- 5.1 Perturbation Error.- 5.2 Stability of Approximation.- 5.3 Reliability of Prediction.- 6 Data Structures and Memory Management.- 6.1 Algorithmic Requirements and Design Criteria.- 6.2 Hashing.- 6.3 Data Structures.- 7 Numerical Experiments.- 7.1 Parameter Studies.- 7.2 Real World Application.- A Plots of Numerical Experiments.- B The Context of Biorthogonal Wavelets.- B.1 General Setting.- B.1.1 Multiscale Basis.- B.1.2 Stable Completion.- B.1.3 Multiscale Transformation.- B.2 Biorthogonal Wavelets of the Box Function.- B.2.1 Haar Wavelets.- B.2.2 Biorthogonal Wavelets on the Real Line.- References.- List of Figures.- List of Tables.- Notation.