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Applied Numerical Methods for Partial Differential Equations

  • Textbook
  • © 2024

Overview

Authors:
  1. Carl L. Gardner

    1. School of Mathematical & Statistical Sciences, Arizona State University, Tempe, USA

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  • Rapidly advances students to the level where they can solve (systems of) nonlinear partial differential equations
  • Provides model programs in MATLAB illustrating each of the numerical methods
  • Emphasizes scientific applications, especially fluid and gas dynamics

Part of the book series: Texts in Applied Mathematics (TAM, volume 78)

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About this book

The aim of this book is to quickly elevate students to a proficiency level where they can solve linear and nonlinear partial differential equations using state-of-the-art numerical methods. It covers numerous topics typically absent in introductory texts on ODEs and PDEs, including:

  • Computing solutions to chaotic dynamical systems with TRBDF2
  • Simulating the nonlinear diffusion equation with TRBDF2
  • Applying Newton’s method and GMRES to the nonlinear Laplace equation
  • Analyzing gas dynamics with WENO3 (1D Riemann problems and 2D supersonic jets)
  • Modeling the drift-diffusion equations with TRBDF2 and PCG
  • Solving the classical hydrodynamic model (electro-gas dynamics) with WENO3 and TRBDF2

The book features 34 original MATLAB programs illustrating each numerical method and includes 93 problems that confirm results discussed in the text and explore new directions. Additionally, it suggests eight semester-long projects.

 

This comprehensive text can serve as the basis for a one-semester graduate course on the numerical solution of partial differential equations, or, with some advanced material omitted, for a one-semester junior/senior or graduate course on the numerical solution of ordinary and partial differential equations. The topics and programs will be of interest to applied mathematicians, engineers, physicists, biologists, chemists, and more.

 

Keywords

Table of contents (9 chapters)

  1. Front Matter

    Pages i-xxii
    Download chapter PDF

  2. Overview

    • Carl L. Gardner
    Pages 1-7

  3. Consistency, Stability, and Convergence

    • Carl L. Gardner
    Pages 9-20

  4. Numerical Methods for ODE IVPs

    • Carl L. Gardner
    Pages 21-48

  5. Numerical Methods for ODE BVPs

    • Carl L. Gardner
    Pages 49-56

  6. Overview of PDEs

    • Carl L. Gardner
    Pages 57-63

  7. Numerical Methods for Parabolic PDEs

    • Carl L. Gardner
    Pages 65-89

  8. Numerical Methods for Elliptic PDEs

    • Carl L. Gardner
    Pages 91-119

  9. Numerical Methods for Hyperbolic PDEs

    • Carl L. Gardner
    Pages 121-183

  10. Numerical Methods for Mixed Type PDEs

    • Carl L. Gardner
    Pages 185-200

  11. Back Matter

    Pages 201-216
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Authors and Affiliations

  • School of Mathematical & Statistical Sciences, Arizona State University, Tempe, USA

    Carl L. Gardner

About the author

Carl Gardner is an Emeritus Professor of Mathematics at Arizona State University, where he taught and did research in Computational Mathematics for 30 years.  Previously he held positions at Bowdoin College, NYU, and Duke University. Professor Gardner's research focuses on computational and theoretical fluid dynamics and the numerical solution of nonlinear partial differential equations.  His primary application areas are charge transport in quantum semiconductor devices, ion transport in biological cells (modeling ionic channels as well as synapses), and supersonic flows in astrophysical jets (modeling interactions of jets with their environments and star formation).  These problems are governed by coupled systems of nonlinear partial differential equations, and exhibit complex fluid dynamical phenomena involving nonlinear wave interactions.

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