AMATH 581

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Role
Scientific Computing (SLN 10234, MWF 8:30-9:20, Lowe 216)
Instruction
Professor J. Nathan Kutz
kutz (at) amath.washington.edu
206-685-3029, Guggenheim Hall 414b
Office Hours: T 3:30-5 and Th 7:30-9 in Gug 414B (EDGE call: 206-685-3029)
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Teaching Assistant: Chris Jones
christopher.r.jones (at) gmail.com
Office Hours: T 1-2:50, Th 10-11:50, F 9:30-11:20 in Gug 406
EDGE Office Hours: W 9:30-10:30, Th 1-2 call (206) 685-8069 or SKYPE: cjones_uw_amath

Lectures and Homework
Video Lectures: EDGE
Course Notes: amath581
Homework: HW 1 (Due 10/23), HW 2 (Due 11/4), HW 3 (Due 11/13), HW 4 (Due 12/2), HW 5 (Due 12/09)
MATLAB: Student Edition (recommended if you do not have access)
Facebook class page: amath581facebook

Prerequisites
Solid background in ODEs and familiarity with PDEs and MATLAB, or permission.
Course Description
Survey of practical numerical solution techniques for ordinary and partial differential equations. Emphasis will be on the implementation of numerical schemes to practical problems of the engineering and physical sciences. Methods for partial differential equations will include finite difference, finite element and spectral techniques. Full use will be made of MATLAB and its built in programming and solving functionality.
Objectives

How to recognize and solve numerically practical problems which may arise in your research. We will solve some serious problems using the full power of MATLAB's built in functions and routines. This class is geared for those who need to get the basics in scientific computing. All major types of PDEs (parabolic, elliptic, and hyperbolic) will be considered in 1D, 2D and 3D in problems ranging from quantum mechanics to fluid flows.

NOTE: This course is a survey of computational methods. The focus is on the implemention of numerical schemes with significant aid from built-in MATLAB functionality such as FFTs, fast matrix solvers, etc. It is not a course in numerical analysis since our coverage of many technical issues is only cursory. A much more comprehensive and detailed treatment of some of the methods covered here is given in AMATH 584, 585, 586.



Figure

Dynamics of a repulsive Bose-Einstein condensate trapped in a 3-D lattice potential. The equation was solved using a filtered spectral method in space and 4th-order Runge-Kutta in time. By the end of this course, you should be able to perform this numerical simulation.

Textbook & Notes

There will be no text for this course. I will provide my notes on-line for you to download. I have several texts which will be on reserve at the library to look through the different sections.
Lecture Notes: .pdf

Reference Texts on Reserve:

1. R. L. Burden and J. D. Faires, Numerical Analysis (Sixth Edition). Brooks/Cole, 1997.
2. L. N. Trefethen, Finite Difference and Spectral Methods. (freely available).
3. L. N. Trefethen, Spectral Methods in MATLAB. SIAM.
4. L. N. Trefethen and D. Bau, Numerical Linear Algebra. SIAM.
5. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. CRC Press.
6. D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer.

Syllabus

(1) Solution Methods for Differential Equations: (2 weeks)

We will begin with ODE solvers applied to both initial and boundary value problems. Our application will be to finding the eigenstates of a quantum mechanical problem or of an optical waveguide.

(a) Initial value problems
(b) Euler method, 2nd- and 4th-order Runge-Kutta, Adams-Bashford
(c) Stability and time stepping issues
(d) Boundary values problems: shooting/collocation/relaxation

(2) Finite Difference Schemes for Partial Differential Equations: (3 weeks)

We will introduce the idea of finite-differencing of differential operators. Our application will be to two problems: vibrating modes of a drum and the evolution of potential vorticity in an advection-diffusion problem of fluid mechanics.

(a) Collocation
(b) Stability and CFL conditions
(c) Time and space stepping routines
(d) Tri-diagonal matrix operations

(3) Spectral Methods for Partial Differential Equations: (3 weeks)

Transform methods for PDEs will be introduced with special emphasis given to the Fast-Fourier Transform. We will revisit the potential vorticity in an advection-diffusion problem of fluid mechanics by using these spectral techniques.

(a) The Fast-Fourier transform (FFT)
(b) Chebychev transforms
(c) Time and space stepping routines
(d) Numerical filtering algorithms

(4) Finite Element Schemes for Partial Differential Equations: (2 weeks)

For complicated computational domains, the use of a finite element scheme is compulsory. The steady-state flow of a fluid over various airfoils will be considered.

(a) Mesh generation
(b) Advanced matrix operations
(c) Boundary conditions

Grading

Your course grade will be determined entirely from your homework. There will be no exams. Each of the five homework sets will be part of a computational notebook generated by the student.

On the due date of each homework, the final homework must be uploaded to SCORELATOR for grading. SCORELATOR will give you up to eight chances to get the results correct. The grade for that homework will be based upon the percentage you have exactly right (compared to my master key). The correctness of your codes will determine 60% of your grade, the remaining 40% will be determined by your final report. Each homework is part of a larger computational notebook that will be submitted at the end of quater. Each homework should be written as if it were an article/tutorial being prepared for submission. I expect a high level of professionalism on these reports. The following is the expected format for homework submission:

Title/author/abstract Title, author/address lines, and short (100 words or less) abstract. (It is not to be a separate title page!)
Sec. I. Introduction and Overview
Sec. II. Theoretical Background
Sec. III. Algorithm Implementation and Development
Sec. IV. Computational Results
Sec. V. Summary and Conclusions
Appendix A MATLAB functions used and brief implementation explanation
Appendix B MATLAB codes
Appendix C (optional) Any algebraically intense calculations (long and drawn out calculations have no business in Sec. II!)

I will grade based upon how completely you solved the homework as well as neatness and little things like: did you label your graphs and include figure captions. I expect the final project to have a brief overview of the chapters along with conluding remarks. Further, the professionalism of the final document will be evaluated.

A few things should be kept in mind when generating your reports:

1. Use a professional grade word processor (Latex or MSword, for example)
2. For equations: Latex already does a nice job, but in Word, use Microsoft Equation Editor 3.0
3. Label your graphs. Include brief figure captions. Reference the figure in the text with a more detailed account of the figure.
4. Figures should be set flush with the top or bottom of a page.
5. Label all equations.
6. Provide references where appropriate.
7. All coding should be shuffled to Appendix A and B. Reference it when necessary.
8. Always remember: this report is being written for YOU! So be clear and concise.
9. Spellcheck.

Department of Applied Mathematics University of Washington, Guggenheim Hall #414, Box 352420, Seattle, WA 98195-2420 USA,
Email 'info' (at amath.washington.edu) Phone 206-543-5493 Fax 206-685-1440