M676: Introduction to Numerical Methods for Partial Differential
Equations
Spring 2008 Reference
Number: 13745;
Instructor: Dr. I. Oprea http://www.math.colostate.edu/~juliana
MWF
11:00 - 11:50, Room: Engineering E203
Overview
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
Coursework
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.
Prerequisites
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.
Textbooks
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