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

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