AMATH 481/581

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Role
Scientific Computing (MWF 8:30-9:20, Lowe 216)
Instruction
Professor J. Nathan Kutz
kutz (at) uw.edu
206-685-3029, Guggenheim Hall 414b
Office Hours: W 9:30-11am (EDGE call: 206-685-3029)
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Teaching Assistant: Pedro Maia
pmaia (at) u.washington.edu
Office Hours: Allen Atrium MWF 3-5pm
EDGE Office Hours (call or skype): MW 1-3pm (206) 685-8069 or skype name "theamath581ta"

Lectures and Homework
Video Lectures: EDGE (online) , On Campus Student
Course Notes: amath581
Discussion Board: Catalyst
Homework: Practice 1 (Due midnight 10/6), HW 1 (Due 10/14), HW 2 (Due 10/28), HW 3 (Due 11/4), HW 4 (Due 11/11), HW 5 (Due 11/22), HW 6 (Due 12/7)
MATLAB: Student Edition (recommended if you do not have access)
SCORELATOR: www.scorelator.com


MATLAB on campus
There is MATLAB access at the ICL lab on campus. You can access this also remotely by following the links to "terminal server".

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.

On or before the due date of each homework, the final homework must be uploaded to SCORELATOR for grading. SCORELATOR will give you up to five 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 100% of your grade.

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