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Mimetic Discretization Methods

By Jose E. Castillo, Guillermo F. Miranda

Chapman and Hall/CRC – 2013 – 260 pages

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    978-1-46-651343-3
    January 10th 2013

Description

To help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous identities and theorems found in vector calculus. Mimetic Discretization Methods focuses on the recent mimetic discretization method co-developed by the first author. Based on the Castillo-Grone operators, this simple mimetic discretization method is invariably valid for spatial dimensions no greater than three. The book also presents a numerical method for obtaining corresponding discrete operators that mimic the continuum differential and flux-integral operators, enabling the same order of accuracy in the interior as well as the domain boundary.

After an overview of various mimetic approaches and applications, the text discusses the use of continuum mathematical models as a way to motivate the natural use of mimetic methods. The authors also offer basic numerical analysis material, making the book suitable for a course on numerical methods for solving PDEs. The authors cover mimetic differential operators in one, two, and three dimensions and provide a thorough introduction to object-oriented programming and C++. In addition, they describe how their mimetic methods toolkit (MTK)—available online—can be used for the computational implementation of mimetic discretization methods. The text concludes with the application of mimetic methods to structured nonuniform meshes as well as several case studies.

Compiling the authors’ many concepts and results developed over the years, this book shows how to obtain a robust numerical solution of PDEs using the mimetic discretization approach. It also helps readers compare alternative methods in the literature.

Contents

Introduction

Continuum Mathematical Models

Physically Motivated Mathematical Concepts and Theorems

General 3-D Use of Flux Vector Densities

Illustrative Examples of PDEs

A Comment on the Numerical Treatment of the grad Operator

Notes on Numerical Analysis

Computational Errors

Order of Accuracy

Norms and Condition Numbers

Linear Systems of Equations

Solution of Nonlinear Equations

Mimetic Differential Operators

Castillo-Grone Method for 1-D Uniform Staggered Grids

Higher-Dimensional CGM

2-D Staggerings

3-D Staggerings

Gradient Compositions

Nullity Tests

Higher-Order Operators

Formulation of Nonlinear and Time-Dependent Problems

Object-Oriented Programming and C++

From Structured to Object-Oriented Programming

Fundamental Concepts in Object-Oriented Programming

Object-Oriented Modeling and UML

Inheritance and Polymorphism

Mimetic Methods Toolkit (MTK)

MTK Usage Philosophy

Study of a Diffusive-Reactive Process Using the MTK

Collaborative Development of the MTK: Flavors and Concerns

Downloading the MTK

Nonuniform Structured Meshes

Divergence Operator

Gradient Operator

Case Studies

Porous Media Flow and Reservoir Simulation

Modeling Carbon Dioxide Geologic Sequestration

Maxwell's Equations

Wave Propagation

Geophysical Flow

Appendix A: Heuristic Deduction of the Extended Form of Gauss' Divergence Theorem

Appendix B: Tensor Concept: An Intuitive Approach

Appendix C Total Force Due to Pressure Gradients

Appendix D: Heuristic Deduction of Stokes' Formula

Appendix E: Curl in a Rotating Incompressible Inviscid Liquid

Appendix F: Curl in Poiseuille’s Flow

Appendix G: Green's Identities

Appendix H: Fluid Volumetric Time-Tate of Change

Appendix I: General Formulation of the Flux Concept

Appendix J: Fourth-Order Castillo-Grone Divergence Operators

References

Index

Sample Problems appear at the end of each chapter.

Name: Mimetic Discretization Methods (eBook)Chapman and Hall/CRC 
Description: By Jose E. Castillo, Guillermo F. Miranda. To help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous identities and theorems found in vector calculus. Mimetic Discretization Methods focuses on the...
Categories: Computational Numerical Analysis, Differential Equations, Mathematical Physics