1st Edition

Iterative Splitting Methods for Differential Equations

By Juergen Geiser Copyright 2011
    320 Pages 71 B/W Illustrations
    by Chapman & Hall

    320 Pages 71 B/W Illustrations
    by Chapman & Hall

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    Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations.

    In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential equations and spatial- and time-dependent differential equations.

    The practical part of the text applies the methods to benchmark and real-life problems, such as waste disposal, elastics wave propagation, and complex flow phenomena. The book also examines the benefits of equation decomposition. It concludes with a discussion on several useful software packages, including r3t and FIDOS.

    Covering a wide range of theoretical and practical issues in multiphysics and multiscale problems, this book explores the benefits of using iterative splitting schemes to solve physical problems. It illustrates how iterative operator splitting methods are excellent decomposition methods for obtaining higher-order accuracy.


    Model Problems
    Related Models for Decomposition
    Examples in Real-Life Applications

    Iterative Decomposition of Ordinary Differential Equations
    Historical Overview
    Decomposition Ideas
    Introduction to Classical Splitting Methods
    Iterative Splitting Method
    Consistency Analysis of the Iterative Splitting Method
    Stability Analysis of the Iterative Splitting Method for Bounded Operators

    Decomposition Methods for Partial Differential Equations
    Iterative Schemes for Unbounded Operators

    Computation of the Iterative Splitting Methods: Algorithmic Part
    Exponential Runge-Kutta Methods to Compute Iterative Splitting Schemes
    Matrix Exponentials to Compute Iterative Splitting Schemes

    Extensions of Iterative Splitting Schemes
    Embedded Spatial Discretization Methods
    Domain Decomposition Methods Based on Iterative Operator Splitting Methods
    Successive Approximation for Time-Dependent Operators

    Numerical Experiments
    Benchmark Problems 1: Introduction
    Benchmark Problems 2: Comparison with Standard Splitting Methods
    Benchmark Problems 3: Extensions to Iterative Splitting Methods
    Real-Life Applications
    Conclusion to Numerical Experiments: Discussion of Some Delicate Problems

    Summary and Perspectives

    Software Tools
    Software Package Unstructured Grids
    Software Package r3t
    Solving PDEs Using FIDOS





    Juergen Geiser is a researcher in the Department of Mathematics at the Humboldt-University of Berlin. His research interests include numerical and computational analysis, partial differential equations, decomposition and discretization methods for hyperbolic and parabolic equations, optimization, scientific computing, and interface analysis.