M550:
Introduction to Numerical Methods for
Partial Differential
Equations
Spring 2012 Reference
Number:
CRN
14828
Instructor: Dr. I. Oprea http://www.math.colostate.edu/~juliana
MWF 8-8:50AM, Room: Weber 201(see comments below)
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. Applications and examples will be correlated to students
interests.
Schedule - MWF
8-8:50AM, but can be adjusted to fit everyone's' schedule
Coursework
The course work
consists
of a mixture of mathematical problem sets and a few
computational projects.
The computational projects emphasize experimentation with
mathematical
issues such as convergence, stability and accuracy. Grading is
based on HW/Projects - 70% and a Take-Home Final Exam-30%.
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.
Recommended
Textbook
Jichun Li; Yi-Tung Chen, University of Nevada,
Las Vegas, - Computational Partial Differential Equations
Using MATLAB
, Series: Chapman & Hall/CRC Applied Mathematics &
Nonlinear Science; Chapman and Hall/CRC -
Supplementary Lecture:
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