M550: Introduction to Numerical Methods for Partial Differential Equations
Spring 2010  Reference Number: CRN 14828
Instructor: Dr. I. Oprea
MWF 12:00 - 12:50, Room: Engineering E206

Analytic solutions exist only for the most elementary partial differential equations (PDEs); the rest must be tackled with numerical methods.  This course will cover numerical solution of PDEs: the method of lines, finite differences, finite element and spectral methods, to an extent necessary for successful numerical modeling of physical phenomena. We shall construct and characterize the behavior of computational methods for PDEs based on the physical meaning and the mathematical analysis of the underlying continuous equations

This course is an entry-level graduate course. The intended audience is graduate students and advanced undergraduates. The design philosophy of the course is to cover the mathematical theory applied to the simplest examples so as to minimize technical details, yet provide a strong foundation that students can carry back to their particular fields

The course work consists of a mixture of mathematical problem sets and computational projects. The computational projects emphasize experimentation with mathematical issues such as convergence, stability and accuracy.

The course assumes familiarity with basic linear algebra and ordinary differential equations. Exposure to solutions of the classic models in partial differential equations is a plus.  The use of MATLAB is strongly encouraged.

    1. Computational Differential Equations, K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Cambridge University Press, 1996
2. Numerical Solution of Partial Differential Equations: An Introduction, K. Morton and D. Mayers, Cambridge University Press, 2-nd edition

Topics to be Covered
1. Brief Introduction to Partial Differential Equations and Basic Numerical Analysis
    - Interpolation theory Numerical quadrature, The need for numerical solutions of differential equations

2. Spectral Methods

    - an overview

2.Elliptic Problems and the Finite Element Method
    - Models involving conservation of heat, behavior of solutions
    - Two-point boundary value problems and the Laplace and Poisson equations
    - The variational formulation and the Galerkin finite element method
    - Convergence in the energy norm, a priori convergence, order of convergence
    - Quadrature in the finite element method and the finite difference method
    - Brief overview of complications that can occur in realistic models
3. Parabolic Problems and the Method of Lines
    - Explicit and implicit method of lines using finite elements in space and finite differences in time
    - Numerical stability, stiffness and dissipativity, convergence
4. Hyperbolic Problems and the Finite Difference Method
    - The transport equation and wave equations, characteristics and the transport of information, behavior of solutions
    - Finite difference schemes, consistency
    - Stability, dissipativity, dispersion, the CFL condition, convergence
5. Miscellaneous Topics as Time Permits : Error estimation, computational error estimation and adaptive schemes, conservation laws