632 pages 9x6 inches w/ CDROM
Jan 2003 Hardcover
ISBN 1-58949-029-0
US$82

 

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Presented in an intuitive and highly readable style, this textbook provides a thorough introduction to the principles, methods and art of solving differential equations on the computer by finite-differencing. Topics range from the elements of discretization, interpolation and stability to standard solvers for ordinary and partial differential equations to advanced topics such as symplectic integrators, flux-corrected transport and multi-grid.  The coverage is leavened with numerous examples and exercises as well as projects designed to expose the student to the material as it is actually practiced.  To further enhance understanding this text comes with an electronic Mathcad version that makes it fully interactive, e.g., inputs to examples can be changed and the results are instantly re-calculated.  The software for a PC is on the accompanying CD-ROM (Mathcad not included). More detail about Mathcad© may be found at http://www.mathsoft.com.


Advanced undergraduates and beginning graduate students in all areas of applied physics and engineering. Also provides a clear and convenient reference for practicing engineers, teachers and researchers.

 

0
.   Preliminaries
      
A. Preface
       B. Some Comments on Style

I.   Introduction
      
A. Overview
       B. The Art of Computational Sciences

II.   Basic Tools
      
A. Introduction
       B. Essential Concepts
            1. Discrete Representations
            2. Interpolation/Extrapolation
            3. Types of Error
            4. Discretization Error
            5. Algorithmic Error
       C. Interpolation/Extrapolation
            1. Lagrange Interpolation
            2. Piecewise Interpolation
            3. Rational Function Approximation
            4. Fourier and Chebyshev Interprolation
       D. Numerical Differentiation
            1. Introduction to Finite Differences
            2. Low-order Approximation
            3. High-order Approximation
       E. Numerical Integration
            1. General-purpose Quadrature
            2. High Efficiency Quadrature
            3. Stochastic Quadrature
       F. Basic Tools Projects

III.   Ordinary Differential Equations
       A. Overview
            1. Introduction
            2. IVPs verses BVPs
            3.  Summary of ODE-IVP Methods
       B. Initial Values Problems -- Basic Methods and Issues
            1. Eulerís Method
            2. Stability
            3. Backward Euler & Centered Schemes
            4. Handling Implicitness
            5. Simulation Examples
            6. Basic Multi-Step Methods
       C. Initial Value Problems -- One-Step Methods
            1. Runge-Kutta Methods
            2. Bulirsch-Stoer Method
            3. Simulation Examples
            4. Exponential Fitting Methods
       D. Initial Value Problems  -- Multi-Step Methods
            1. Introduction
            2. Adams-Bashforth-Moulton Methods
       E. ODE Systems
            1. Solving ODE Systems
            2. Stiffness
            3. Stiff Bulirsch-Stoer Method
            4. Gearís Methods
            5. Simulation Example

            6. Asymptotic Approach
            7. Oscillatory Behavior
       F. Two-Point Boundary Value Problems
            1. Shooting Methods 
            2. Implicit Methods
       G. ODE Projects

IV.  Partial Differential Equations: Finite-
    Difference Methods
       A. Overview
       B. Parabolic PDEs
            1. Diffusion Equation
            2. A Low-Order Method
            3. Stability
            4. Crank-Nicolson
            5. Compact Methods
            6. Simulation examples: 1-D
            7. Scalar Implicit Methods
            8. Splitting Methods
            9. Simulation examples: 2-D
       C. Elliptic PDEs
            1. Introduction
            2. Relaxation Methods
            3. Simulation Examples
            4. Nonlinear Problems
            5. Multi-Grid Approach
            6. Conjugate-Gradient Method
            7. Fast Direct Methods
            8. Non-Uniform Grids
       D. Hyperbolic PDEs
            1. Wave Equation
            2. Numerical Diffusion and Dispersion
            3. Low-Order Explicit Methods
            4. High-Order Schemes
            5. Flux-Corrected Transport
                Algorithm
            6. Simulation example
            7. Nonlinear Problems
        E. PDE Projects
           
V.  Partial Differential Equations: short introduction to Basis-Function Expansion Methods
       A. Basics Function expansion Methods
       B. Finite Element Method: Elliptic PDEs
            1. Finite Elements in One Dimension      
       C. Spectral Method: Hyperbolic PDEs
            1. Introduction to the Spectral Method

VI.  References

VII.  Appendices
       A. Tridiagonal Systems
             1. Algorithms
             2. Examples
       B. Vector Algebra and Vector Calculus
       C. Input/Output
       D. Project Guidelines



 

Dr. Mario G. Ancona has been a part-time faculty member for 14 years in Applied Physics in the Part-Time Programs in Engineering and Applied Science at Johns Hopkins University, where he has taught classes in Solid State Physics, Electromagnetism and Electron Devices in addition to the course on Computational Physics.  He is also on the staff of the Electronics Science & Technology Division of the Naval Research Laboratory where his primary area of research has been the modeling and simulation of electronic devices, circuits and processing.