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.