The Essentials

Syllabus

• Basic equations of mathematical physics, other application domains
• Minimal residual formulation
• Numerical methods for ODE's
• Linear advection diffusion equations
• Parabolic equations
• Hyperbolic equations
• Equations of mixed type
• Coupled PDE-ODE systems

Motivation and Objectives

The physical world is described by a small set of laws. One of the most useful mathematical statements of these physical laws is in the form of partial differential equations (PDE's) describing local changes in physical parameters. Though it is usually straightforward to write down the particular PDE's that correspond to a given problem, finding a solution is much more difficult. For the vast majority of real-world problems approximate techniques must be used. One of the most productive approaches is to use a numerical approximation of the PDE's of interest. These techniques are used in studying chemical reactions, ecological models, astrophysical phenomena, aircraft design, financial models and in many other application domains.

Grading Policy

Grading shall be determined based on homework (HW, 48 points), supplementary reading (SR, 8 points), final project (FP, 16 points), midterm examination (ME, 14 points) and final examination (FE, 14 points). The number of points accumulated during classwork is mapped to a graduate grade as shown in the following table.

Each interval is 15 points; H+/H/H- correspond to 5 point subintervals and so on.

Course Texts and Notes

One of the tenets of graduate study is the ability to critically examine multiple sources in order to verify theories and to enhance understanding. Lecture notes for the course shall be provided online in PDF format. These are meant to document the progress of material presented in class. It is expected and required that the student supplement the lecture notes with reading from other sources.

General bibliography

The following texts serve as general background material. Students are encouraged to actively read from these texts material related to the current lecture. Homework and final examination questions from this material should be expected.

• Finite Difference Schemes and Partial Differential Equations, John Strikwerda

• Numerical Methods for Conservation Laws, Randall LeVeque (Cambridge)

• Finite Difference Schemes for Computational Fluid Dynamics, Phillip Colella & Gerry Puckett

Topic bibliography

Texts providing more details on specific aspects we shall discuss shall be listed here as the course progresses.

ODE methods

• Finite Difference Methods for Ordinary and Partial Differential Equations, Randall LeVeque (SIAM)

• A First Course in the Numerical Analysis of Differential Equations, Arieh Iserles (Cambridge)

Homework and Projects

Useful C and Fortran coding techniques

• Defining a function to be an argument to another procedure fEuler.c fEuler.f90

Computer codes

Software

The course homework solutions use the following freely available software:

• TeXmacs http://www.texmacs.org

• Python with the SciPy packages http://www.python.org, http://www.scipy.org, especially the PyLab Matlab-like environment http://www.scipy.org/PyLab. An excellent Python distribution for the PC and Mac is offered by Enthought http://www.enthought.com/products/epd.php (free for academic use)

• Gnuplot http://www.gnuplot.info/

• Maxima (symbolic computation package) http://maxima.sourceforge.net/

Examinations

Mid-term examination solution: midterm.pdf

Final examination solution: final.pdf

Final examination is Friday, Dec. 12, noon, Ph301.

Course Bulletin Boards

Please choose the appropriate section for your question. Questions are answered daily, typically late at night.