Mimetic discretization methods / José E. Castillo, Guillermo F. Miranda.

Format:

Book

Author/Creator:

Castillo, José E.

Contributor:

Miranda, Guillermo F.

Class of 1924 Book Fund.

Language:

English

Subjects (All):

Numerical analysis.

Physical Description:

xxiii, 235 pages : illustrations ; 24 cm

Place of Publication:

Boca Raton : CRC Press, [2013]

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. Book jacket.

Contents:

1 Introduction 1

2 Continuum Mathematical Models 7

2.1 Physically Motivated Mathematical Concepts and Theorems 8

2.1.1 Flux and Flux Density 8

2.1.2 Gauss' Divergence Theorem 9

2.1.3 Work, Circulation, and Electromotive Force 11

2.1.4 Stokes' Theorem and curl 12

2.1.5 2-D Green's Formula 15

2.1.6 Laplace's Equation and 2-D Incompressible and Inviscid Flows ("Potential Flows") 15

2.2 General 3-D Use of Flux Vector Densities 21

2.2.1 Important Particular Cases 21

2.3 Illustrative Examples of PDEs 22

2.3.1 The Continuity Equation 22

2.3.2 Maxwell's Equations 24

2.4 A Comment on the Numerical Treatment of the grad Operator 26

2.5 Concluding Remarks 26

2.6 Sample Problems 27

3 Notes on Numerical Analysis 31

3.1 Computational Errors 31

3.2 Order of Accuracy 32

3.3 Norms and Condition Numbers 33

3.3.1 Condition Number of a Matrix 35

3.4 Linear Systems of Equations 35

3.4.1 Direct Methods 36

3.4.2 Iterative Methods 37

3.5 Solution of Nonlinear Equations 38

3.6 Concluding Remarks 39

3.7 Sample Problems 40

4 Mimetic Differential Operators 43

4.1 Castillo-Grone Method for 1-D Uniform Staggered Grids 48

4.2 Higher-Dimensional CGM 56

4.3 2-D Staggerings 58

4.3.1 2-D Gradient 58

4.3.2 2-D Divergence 61

4.3.3 2-D curl 61

4.4 3-D Staggerings 65

4.4.1 3-D Gradient 65

4.4.2 3-D Divergence 66

4.4.3 3-D curl 66

4.5 Gradient Compositions: ∇ ∇ = ∇² 68

4.5.1 Hybrid Nodes and Computing the Laplacian 69

4.6 Nullity Tests 71

4.7 Higher-Order Operators 71

4.7.1 Sixth-Order Operators 71

4.7.2 Eighth-Order Operators 74

4.8 Formulation of Nonlinear and Time-Dependent Problems 77

4.9 Concluding Remarks 77

4.10 Sample Problems 78

5 Object-Oriented Programming and C++ 81

5.1 From Structured to Object-Oriented Programming 81

5.2 Fundamental Concepts in Object-Oriented Programming 84

5.2.1 Access Specification and Implementation Hiding 85

5.2.2 Mutators, Accessors, and Client Code 87

5.3 Object-Oriented Modeling and UML 91

5.3.1 Relationships Among Classes 92

5.3.2 Multiplicity of a Relationship 93

5.4 Inheritance and Polymorphism 93

5.4.1 Polymorphism and Operator Overloading 94

5.5 Concluding Remarks 101

5.6 Sample Problems 101

6 Mimetic Methods Toolkit (MTK) 103

6.1 MTK Usage Philosophy 104

6.2 Study of a Diffusive-Reactive Process Using the MTK 107

6.3 Collaborative Development of the MTK: Flavors and Concerns 116

6.4 Downloading the MTK 118

6.5 Concluding Remarks 118

6.6 Sample Problems 118

7 Nonuniform Structured Meshes l21

7.1 Divergence Operator 121

7.2 Gradient Operator 123

7.3 Concluding Remarks 126

7.4 Sample Problems 126

8 Case Studies 129

8.1 Porous Media Flow and Reservoir Simulation 129

8.1.1 Darcy's Law and the Pressure Equation 130

8.1.2 Diphasic Flow and Permeability 130

8.1.3 Generalization and the Material Tensor 131

8.1.4 Example Using a Mimetic Method 132

8.1.5 Arising Systems of Equations 136

8.1.6 Example Problem and Results 138

8.1.7 Sample Problems 139

8.2 Modeling Carbon Dioxide Geologic Sequestration 141

8.2.1 Introduction 141

8.2.2 The Importance of Studying the Long-Term Behavior of Injected CO₂ 141

8.2.3 Enhanced Oil Recovery and CCUS 142

8.2.4 Water-Rock Interaction and Reactive Transport Modeling and Simulation 143

8.2.5 Potential Applications for Mimetic Discretization Methods 145

8.2.6 Initial Computational Implementation 146

8.2.7 Sample Problems 148

8.3 Maxwell's Equations 149

8.3.1 Background 150

8.3.2 Use of the Castillo-Grone Method 151

8.3.3 A 1-D Example Problem 152

8.3.4 A 2-D Example Problem 153

8.3.5 Sample Problems 154

8.4 Wave Propagation 155

8.4.1 Problem Formulation 156

8.4.2 Methodology 157

8.4.3 Application of the Castillo-Grone Method 160

8.4.4 Attained Results 161

8.4.5 Sample Problems 164

8.5 Geophysical Flow 165

8.5.1 Equation of Motion in the Atmosphere and Oceans 165

8.5.2 Boussinesq Approximation and Incompressible Equations 167

8.5.3 Shallow Water Equations 167

8.5.4 Continuity Equation and Poisson Equation for Preasure 168

8.5.5 Numerical Solution of the Navier-Stokes Equation 169

8.5.6 The Mimetic Approach 171

8.5.7 Sample Problems 173

8.6 Concluding Remarks 175.

Notes:

"A Chapman & Hall book."

Includes bibliographical references (pages 217-230) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1924 Book Fund.

ISBN:

1466513438

9781466513433

OCLC:

768171341

Publisher Number:

99953367565