1st Edition
Decomposition Methods for Differential Equations Theory and Applications
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.
Preface
Introduction
Modeling: Multi-Physics Problems
Introduction
Models for Multi-Physics Problems
Examples for Multi-Physics Problems
Abstract Decomposition and Discretization Methods
Decomposition
Discretization
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
Introduction
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
Notation
Appendix A: Software Tools
Appendix B: Discretization Methods
Literature
References
Index
Biography
Jürgen Geiser is a professor in the Department of Mathematics at Humboldt University of Berlin.
