Applied Numerical Methods for Partial Differential Equations

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.

 

1 Overview.- 2 Consistency, Stability, Convergence.- 3 Numerical Methods for ODE IVPs.- 4 Numerical Methods for ODE BVPs.- 5 Overview of PDEs.- 6 Numerical Methods for Parabolic PDEs.- 7 Numerical Methods for Elliptic PDEs.- 8 Numerical Methods for Hyperbolic PDEs.- 9 Numerical Methods for Mixed Type PDEs.- A Useful Mathematical Formulas.- B Norms and Condition Number.- References.- Index .

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.