Mimetic Discretization Methods

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Features

  • Covers the mimetic discretization method based on Castillo-Grone mimetic differential operators
  • Includes an introduction to object-oriented programming and C++ for readers new to the field
  • Explains how the MTK can solve equations in physical and engineering problems
  • Explores case studies involving porous media flow, carbon dioxide geologic sequestration, electromagnetism, seismic wave propagation, and geophysical fluid flow
  • Provides many problems at the end of each chapter along with additional problems and the MTK API at www.csrc.sdsu.edu/mimetic-book

Summary

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.

Table of 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.