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.
Table of Contents
Related Models for Decomposition
Examples in Real-Life Applications
Iterative Decomposition of Ordinary Differential Equations
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
Benchmark Problems 1: Introduction
Benchmark Problems 2: Comparison with Standard Splitting Methods
Benchmark Problems 3: Extensions to Iterative Splitting Methods
Conclusion to Numerical Experiments: Discussion of Some Delicate Problems
Summary and Perspectives
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.