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Numerical Methods for PDEs (Math 566)


This course will introduce you to methods for solving partial differential equations (PDEs) using finite difference methods. The course content is roughly as follows :

The course will mostly focus on 1d , but if there is interest, we can also look at examples of 2d problems. The course will have roughly 5 homework projects and possible midterm. In lieu of a final, students will do a final project which they will present in class.

Send me an e-mail

Please send me an e-mail at donnacalhoun@boisestate.edu so that I can compile an e-mail list for the class. At the very least, include a subject header that says "Math 566". You may leave the message area blank, if you wish, or send me a short note about what you hope to get out of this course.


Basic course information

Instructor Prof. Donna Calhoun
Time Monday/Wednesday 3:00-4:15
Place MG 124
Office Hours TBA
Prerequesites Math 465/565 or permission of instructor

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Required and suggested textbooks

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Lectures

Week #1 (Jan. 22)
Wednesday --  Three basic PDE types

Week #2 (Jan. 27)
Monday --  Chapter 1 : Finite Difference Approximations (Sec. 1.1-1.3)
Wednesday --  Chapter 1 : Finite Difference Approximations (Sec. 1.4-1.5)
Codes written or demonstrated in class :
fd.m

Week #3 (Feb. 3)
Monday --  Chapter 2 : Steady State and Boundary Value Problems (Sec. 2.1-2.3)
Wednesday --  Chapter 2 : Steady State and Boundary Value Problems (Sec. 2.4-2.6)

Week #4 (Feb. 10)
Monday --  Chapter 2 : Steady State and Boundary Value Problems (Sec. 2.7-2.9)
Some useful or interesting links
The now famous videos (Physics Buzz : 1+2+3+4+5+... = -1/12 !!!)
The Wikipedia page (Wikipedia : 1+2+3+4+5)
Terence Tao's Blog page (Tao Blog)
Wednesday --  Chapter 3 : Elliptic Equations (Sec. 3.1-3.4)

Week #5 (Feb. 17)
Monday --  President's Day
Wednesday --  Chapter 4 : Iterative Methods for sparse Linear Systems (Sec. 4.1-4.2

Week #6 (Feb. 24)
Monday --  Chapter 4 : Iterative Methods for Sparse Linear Systems (Sec. 4.3-4.4)
Wednesday --  Chapter 5 : Initial Value Problems (Sec. 5.1-5.3)

Week #7 (Mar. 3)
Monday --  Chapter 5 : Initial Value Problems (Sec. 5.4-5.6)
Wednesday --  Chapter 5 : Initial Value Problems (Sec. 5.7-5.9)

Week #8 (Mar. 10)
Monday --  Chapter 6 : Zero-stability and Convergence for Initial Value Problems (Sec. 6.1-6.2)
Codes written or demonstrated in class :
tst_sd_sample.m
Wednesday --  Chapter 6 : Stability of Euler's Method (Sec. 6.3-6.4)

Week #9 (Mar. 17)
Monday --  Chapter 6: Stabilty of non-linear methods; connection to BVP (Section 6.3)
Wednesday --  Review of linear multi-step methods; Zero-stability of LMM (Section 5.9; 6.4)
Codes written or demonstrated in class :
lmm_test.m

Week #10 (Mar. 31 )
Monday --  Chapter 7 : Absolute Stability for Ordinary Differential Equations (Sec. 7.1-7.3)
Wednesday --  Chapter 7 : Absolute Stability for Ordinary Differential Equations (Sec. 7.4-7.6)

Week #11 (Apr. 7)
Monday --  Chapter 8 : Stiff Ordinary Differential Equations (Sec. 8.1-8.3)
Wednesday --  Chapter 8 : Stiff Ordinary Differential Equations (Sec. 8.4-8.6)

Week #12 (Apr. 14)
Monday --  Chapter 9 : Diffusion Equations and Parabolic Problems (Sec. 9.1-9.4)
Wednesday --  Chapter 9 : Diffusion Equations and Parabolic Problems (Sec. 9.5-9.7)

Week #13 (Apr. 21)
Monday --  Solving parabolic equations; L-stability
Codes written or demonstrated in class :
solve_heat.m
Wednesday --  Chapter 10 : Advection Equations and Hyperbolic Systems (Sec. 10.1-10.2)
Codes written or demonstrated in class :
solve_heat3.m

Week #14 (Apr. 28)
Monday --  Chapter 10 : Advection Equations and Hyperbolic Systems (Sec. 10.3)
Wednesday --  Chapter 10 : Advection Equations and Hyperbolic Systems (Sec. 10.3-10.4)

Week #15 (May 5)
Monday --  Chapter 10 : Advection Equations and Hyperbolic Systems (Sec. 10.4-10.5)
Codes written or demonstrated in class :
hyperstab.m
Wednesday --  Chapter 11 : Mixed Equations (Sec. 11.1-11.3)

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Homework Assignments

Homework projects are designed to enforce mathematical concepts discussed in class.

Homework #1

Due Feb. 26

Assignment :
hmwk1.pdf
Solutions :
prob1_solns.pdf  -- Matlab solutions to problem 1
prob2_solns.pdf  -- Matlab solutions to problem 2
prob2_page1.jpg  -- Handwritten solutions to problem 2
prob2_page2.jpg  -- Handwritten solutions to problem 2
prob2_page3.jpg  -- Handwritten solutions to problem 2
prob3_solns.pdf  -- Matlab solutions to problem 3
prob3_solns.jpg  -- Handwritten solutions to problem 3

Homework #2

Due April 4th

Assignment :
hmwk2.pdf

Homework #3

Due April 9th

Assignment :
hmwk3.pdf

Homework #4

Due April 28th

Assignment :
hmwk4.pdf
Comments : I may still change minor details on problem 4 (problem size; initial conditions), so please check back for the final version.

Homework #5

Due May 9th

Assignment :
hmwk5.pdf
Other resources :
hyper_demo.m
wave_eqn.m

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Final Project

In lieu of a final, you will have a final project due, on the day of our scheduled exam. Please turn in a 1-2 page project proposal by the end of week 10. A 5-10 page write-up of your project will be due during the week of finals.

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Matlab tutorials

Below are a series of tutorials that should help you become familar with Matlab syntax

  1. Using Matlab as a Graphing Calculator, Part I
  2. Using Matlab as a Graphing Calculator, Part II
  3. Using Matlab as a Graphing Calculator, Part III
  4. Loops and one dimensional arrays
  5. Two dimensional arrays and introduction to matrices
  6. Vandermonde matrix systems
  7. Lagrange polynomials

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Grading policy

Homeworks will count for 75% of your final grade, and a final project will count towards 25% of your final grade.

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