1st Edition

Decomposition Methods for Differential Equations Theory and Applications

By Juergen Geiser Copyright 2009
    304 Pages 43 B/W Illustrations
    by CRC Press

    304 Pages 43 B/W Illustrations
    by CRC Press

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    Decomposition Methods for Differential Equations: Theory and Applications describes the analysis of numerical methods for evolution equations based on temporal and spatial decomposition methods. It covers real-life problems, the underlying decomposition and discretization, the stability and consistency analysis of the decomposition methods, and numerical results.

    The book focuses on the modeling of selected multi-physics problems, before introducing decomposition analysis. It presents time and space discretization, temporal decomposition, and the combination of time and spatial decomposition methods for parabolic and hyperbolic equations. The author then applies these methods to numerical problems, including test examples and real-world problems in physical and engineering applications. For the computational results, he uses various software tools, such as MATLAB®, R3T, WIAS-HiTNIHS, and OPERA-SPLITT.

    Exploring iterative operator-splitting methods, this book shows how to use higher-order discretization methods to solve differential equations. It discusses decomposition methods and their effectiveness, combination possibility with discretization methods, multi-scaling possibilities, and stability to initial and boundary values problems.



    Modeling: Multi-Physics Problems


    Models for Multi-Physics Problems

    Examples for Multi-Physics Problems

    Abstract Decomposition and Discretization Methods



    Time-Decomposition Methods for Parabolic Equations

    Introduction for the Splitting Methods

    Iterative Operator-Splitting Methods for Bounded Operators

    Iterative Operator-Splitting Methods for Unbounded Operators

    Decomposition Methods for Hyperbolic Equations

    Introduction for the Splitting Methods

    ADI Methods and LOD Methods

    Iterative Operator-Splitting Methods for Wave Equations

    Parallelization of Time Decomposition Methods

    Nonlinear Iterative Operator-Splitting Methods

    Spatial Decomposition Methods

    Domain Decomposition Methods Based on Iterative Operator-Splitting Methods

    Schwarz Waveform-Relaxation Methods

    Overlapping Schwarz Waveform Relaxation for the Solution of Convection-Diffusion-Reaction Equation

    Numerical Experiments


    Benchmark Problems for the Time Decomposition Methods for Ordinary Differential and Parabolic Equations

    Benchmark Problems for Spatial Decomposition Methods: Schwarz Waveform-Relaxation Method

    Benchmark Problems: Hyperbolic Equations

    Real-Life Applications

    Summary and Perspectives


    Appendix A: Software Tools

    Appendix B: Discretization Methods





    Jürgen Geiser is a professor in the Department of Mathematics at Humboldt University of Berlin.