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    Mimetic Discretization Methods

    ISBN-10: 1466513438
    ISBN-13: 9781466513433
    Author(s): Jose E. Castillo, Guillermo F. Miranda
    Description: There are many limitations in the current methods to solve partial differential equations and mimetics method have been demonstrated in many applications to be produce results consisting with the physics of the problem under investigation. This book  More...
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    Publisher: CRC Press
    Binding: Hardcover
    Pages: 260
    Size: 6.50" wide x 9.50" long x 1.00" tall
    Weight: 1.100
    Language: English

    There are many limitations in the current methods to solve partial differential equations and mimetics method have been demonstrated in many applications to be produce results consisting with the physics of the problem under investigation. This book provides a robust and general method to solve partial differential equations. Written for graduate students in science and engineering, the book also suitable for advanced undergraduates. It will be a reference for any researcher working in any area where solving PDE’s is required. It could also be use as a textbook.

    List of Figures
    List of Tables
    List of Algorithms
    List of Symbols
    Preface
    Acknowledgments
    Introduction
    Continuum Mathematical Models
    Physically Motivated Mathematical Concepts and Theorems
    Flux and Flux Density
    Gauss' Divergence Theorem
    Work, Circulation, and Electromotive Force
    Stokes' Theorem and curl
    2-D Green's Formula
    Laplace's Equation and 2-D Incompressible and Inviscid Flows ("Potential Flows")
    General 3-D Use of Flux Vector Densities
    Important Particular Cases
    Illustrative Examples of PDEs
    The Continuity Equation
    Maxwell's Equations
    A Comment on the Numerical Treatment of the grad Operator
    Concluding Remarks
    Sample Problems
    Notes on Numerical Analysis
    Computational Errors
    Order of Accuracy
    Norms and Condition Numbers
    Condition Number of a Matrix
    Linear Systems of Equations
    Direct Methods
    Iterative Methods
    Solution of Nonlinear Equations
    Concluding Remarks
    Sample Problems
    Mimetic Differential Operators
    Castillo-Grone Method for 1-D Uniform Staggered Grids
    Higher-Dimensional CGM
    2-D Staggerings
    2-D Gradient
    2-D Divergence
    2-D curl
    3-D Staggerings
    3-D Gradient
    3-D Divergence
    3-D curl
    Gradient Compositions: &#8711; &#8711; = &#8711;<sup>2</sup>
    Hybrid Nodes and Computing the Laplacian
    Nullity Tests
    Higher-Order Operators
    Sixth-Order Operators
    Eighth-Order Operators
    Formulation of Nonlinear and Time-Dependent Problems
    Concluding Remarks
    Sample Problems
    Object-Oriented Programming and C++
    From Structured to Object-Oriented Programming
    Fundamental Concepts in Object-Oriented Programming
    Access Specification and Implementation Hiding
    Mutators, Accessors, and Client Code
    Object-Oriented Modeling and UML
    Relationships Among Classes
    Multiplicity of a Relationship
    Inheritance and Polymorphism
    Polymorphism and Operator Overloading
    Concluding Remarks
    Sample Problems
    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
    Concluding Remarks
    Sample Problems
    Nonuniform Structured Meshes l21
    Divergence Operator
    Gradient Operator
    Concluding Remarks
    Sample Problems
    Case Studies
    Porous Media Flow and Reservoir Simulation
    Darcy's Law and the Pressure Equation
    Diphasic Flow and Permeability
    Generalization and the Material Tensor
    Example Using a Mimetic Method
    Arising Systems of Equations
    Example Problem and Results
    Sample Problems
    Modeling Carbon Dioxide Geologic Sequestration
    Introduction
    The Importance of Studying the Long-Term Behavior of Injected CO<sub>2</sub>
    Enhanced Oil Recovery and CCUS
    Water-Rock Interaction and Reactive Transport Modeling and Simulation
    Potential Applications for Mimetic Discretization Methods
    Initial Computational Implementation
    Sample Problems
    Maxwell's Equations
    Background
    Use of the Castillo-Grone Method
    A 1-D Example Problem
    A 2-D Example Problem
    Sample Problems
    Wave Propagation
    Problem Formulation
    Methodology
    Application of the Castillo-Grone Method
    Attained Results
    Sample Problems
    Geophysical Flow
    Equation of Motion in the Atmosphere and Oceans
    Boussinesq Approximation and Incompressible Equations
    Shallow Water Equations
    Continuity Equation and Poisson Equation for Preasure
    Numerical Solution of the Navier-Stokes Equation
    The Mimetic Approach
    Sample Problems
    Concluding Remarks
    Heuristic Deduction of the Extended Form of Gauss' Divergence Theorem
    Standard Form of Gauss' Divergence Theorem
    Extended Form of Gauss' Divergence Theorem
    Tensor Concept: An Intuitive Approach
    Newton's Law Applied to the Mass in the Tetrahedron
    Newton's Law on a Parallelepiped
    Total Force Due to Pressure Gradients
    Heuristic Deduction of Stokes' Formula
    Curl in a Rotating Incompressible Inviscid Liquid
    Curl in Poiseuille's Flow
    Green's Identities
    Fluid Volumetric Time-Tate of Change
    General Formulation of the Flux Concept
    Fourth-Order Castillo-Grone Divergence Operators
    Computing the Weight Matrix Q
    References
    Index

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