Iterative Splitting Methods for Differential Equations

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.

Introduction Model ProblemsRelated Models for Decomposition Examples in Real-Life Applications Iterative Decomposition of Ordinary Differential EquationsHistorical Overview Decomposition IdeasIntroduction to Classical Splitting MethodsIterative Splitting Method Consistency Analysis of the Iterative Splitting MethodStability Analysis of the Iterative Splitting Method for Bounded Operators Decomposition Methods for Partial Differential EquationsIterative Schemes for Unbounded Operators Computation of the Iterative Splitting Methods: Algorithmic PartExponential Runge-Kutta Methods to Compute Iterative Splitting Schemes Matrix Exponentials to Compute Iterative Splitting Schemes Algorithms Extensions of Iterative Splitting Schemes Embedded Spatial Discretization MethodsDomain Decomposition Methods Based on Iterative Operator Splitting MethodsSuccessive Approximation for Time-Dependent Operators Numerical ExperimentsIntroduction Benchmark Problems 1: IntroductionBenchmark Problems 2: Comparison with Standard Splitting MethodsBenchmark Problems 3: Extensions to Iterative Splitting MethodsReal-Life ApplicationsConclusion to Numerical Experiments: Discussion of Some Delicate Problems Summary and Perspectives Software ToolsSoftware Package Unstructured GridsSoftware Package r3tSolving PDEs Using FIDOS Appendix Bibliography Index

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.