Carlos J. García-Cervera.
South Hall, 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Monday, Wednesday, and Friday 12:45-1:45pm

• Finite Difference Schemes and Partial Differential Equations, by John C. Strikwerda.
• Difference Methods for Initial-Value Problems, by Richtmyer Morton.
• A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles.
• Spectral Methods in Fluid Dynamics, by Canuto, Hussaini, Quarteroni and Zang.
• Numerical Analysis of Spectral Methods, by David Gottlieb and Steven A. Orszag.
• Numerical Methods for Conservation Laws, by Randall J. LeVeque.

• Vector calculus, by Jerrold E. Marsden and Anthony J. Tromba.

• Numerical Analysis, by Richard Burden and J. Douglas Faires.

Course description: This is a graduate course in Numerical Analysis, with an emphasis in the solution of Partial Differential Equations using Finite Differences. If time permits, I will also consider Spectral Methods. In this course we will study the numerical solution of elliptic, hyperbolic, and parabolic equations. Although previous knowledge in Numerical Analysis is recommended, it is not a requirement for this course. I will cover briefly the solution of linear systems of equations, with emphasis in Fast Poisson solvers, such as solution by Fast Fourier Transform. During the course I will illustrate the methods studied with applications from different areas, such as Fluid Dynamics and Electromagnetism. Although the emphasis will be in applications, the course will have a strong theoretical component.

Prerequisites: Knowledge of a computer language suitable for numerical computing: FORTRAN, C, C++, or Matlab. Previous knowledge in Numerical Analysis is recommended, although not necessary.

Assignments and grading: Homework will be assigned every two weeks. It will be handed out on Fridays, and will be collected at the beginning of the class on the Friday of the second following week. Late homeworks will not be accepted. The homework will generally consist of some theoretical questions, and some computational assignments. You will be required to write a program to solve certain problems. The program must be given to me as part of the assignment, together with the output of the program, in the format indicated in the assignment, and an interpretation of the results whenever necessary. You can write the programs either in FORTRAN, C, C++, or Matlab.

In addition to the homework, there will be a final project. Your final grade for the course will be decided according to the following formula:

Final Grade = 60% Homework + 40% Final

You should try and do a search on the Internet, since there are a lot of sites dedicated to Matlab, and programming in general.

Professor Douglas Arnold, at the Institute for Mathematics and its Applications, has a list of disaters due to bad numerical computing. Check it out here!

I wrote the syllabus and the homework assignments using a program called LaTeX. If you want to learn more about this program, you can find some information and tutorials at the following homepages:

• The LaTeX Project.
• Text processing using LaTeX.
• An introduction to LaTeX.

Syllabus: During this course we will try to follow the following schedule. However, much like everything said earlier, this is subject to change.

Week 1: Hyperbolic Partial Differential Equations.

1. Overview of Hyperbolic Partial Differential Equations.
2. Boundary Conditions.
3. Introduction to Finite Difference Schemes.
4. Convergence and Consistency.
5. Stability.
6. The Courant-Friedrichs-Lewy Condition.
7. The Lax-Richtmyer Equivalence Theorem.
8. Homework 1: postscript, pdf. Due on Friday, April 18th, 2003.

Week 2: Analysis of Finite Difference Schemes.

1. Fourier Analysis.
2. Von Neumann Analysis.

Weeks 3 & 4: Order of Accuracy of Finite Difference Schemes.

1. Order of Accuracy.
2. Difference Notation and Difference Calculus.
3. Boundary Conditions for Finite Difference Schemes.
4. Solving Tridiagonal Systems.
5. Fast Poisson Solvers: FFT and Multigrid.

Week 5: Stability for Multistep Schemes.

1. Stability for the Leapfrog Scheme.
2. Stability for General Multistep Schemes.
3. Homework 2: postscript, pdf. Due on Friday, May 2nd, 2003.

Weeks 6 & 7: Dissipation and Dispersion.

1. Dissipation.
2. Dispersion.
3. Group Velocity and the Propagation of Wave Packets.

Week 8: Parabolic Partial Differential Equations.

1. Overview of Parabolic Differential Equations.
2. Parabolic Systems and Boundary Conditions.
3. Finite Difference Schemes for Parabolic Equations.
4. The Convection-Diffusion equation.
5. Homework 3: postscript, pdf. Due on Monday, May 19th, 2003.

Week 9: Systems of Partial Differential Equations.

1. Stability of Finite Difference Schemes for Systems.
2. Finite Difference Schemes in Two and Three Dimensions.

Weeks 10 & 11: Spectral Methods for Partial Differential Equations.

1. Spectral Approximation: The Fourier System, Orthogonal Polynomials in [-1,1], and Chebyshev Polynomials.
2. Fourier Galerkin.
3. Fourier Collocation.
4. Chebyshev Tau.
5. Chebyshev Collocation.
6. Temporal Discretization.