Math 566
Numerical Analysis II
Spring 2015

Basic Information

Time and Place:  TuTh 9:00-10:15am, Mathematics Building 124
Instructor:  Dr. Grady Wright, MB 140, 426-4674, Email:
Office Hours:  TuTh: 12:30-1:45pm, or by appointment (please e-mail to set it up).

Text Books

Lectures


 
Day Topics Material
13-Jan-2015
Numerical differentiation
Ch. 1.1-1.5
15-Jan-2015
Numerical differentiation
Example
Fornberg's algorithm for calculating finite-difference formulas
Ch. 1.1-1.5
20-Jan-2015
Numerical differentiation
Trigonometric interpolation
22-Jan-2015
Lecture
Trigonometric interpolation
Discrete Fourier Transform
27-Jan-2015
Lecture
Discrete Fourier Transform
29-Jan-2015
Lecture
Fast Fourier Transform
03-Feb-2015
Lecture
Discrete convolution theorem
Sine/Cosine transform
05-Feb-2015
Lecture
Boundary value problems: finite difference method
Ch. 2.1-2.7
10-Feb-2015
Lecture
Boundary value problems: finite difference method
Ch. 2.8-2.13
12-Feb-2015
Lecture
Boundary value problems: finite difference method
Ch. 2.13-2.15
17-Feb-2015
Lecture
Higher order methods
Collocation methods
Ch. 2.20-2.21
19-Feb-2015
Lecture
Pseudospectral methods
Elliptic equations
 
Ch. 2.21
Ch. 3.1-3.3
24-Feb-2015
Lecture
Elliptic equations (handout)
 
Ch. 3.3-3.7
26-Feb-2015
Lecture
Direct Poisson solvers (handout)
 
Ch. 3.7
03-Mar-2015
Lecture
Iterative methods for Poisson's equation
 
Ch. 4.1-4.2
05-Mar-2015
Lecture
Initial value problems
Existence and uniqueness
Euler's method
 
Ch. 5.1-5.2, 6.1-6.3
10-Mar-2015
Lecture
Derivations of Euler's method and improvements
Convergence of Euler's method
Stencils
 
12-Mar-2015
Lecture
Local truncation error
Characteristic polynomials
 
19-Mar-2015
Lecture
Zero stability
Order of accuracy
Dahlquist equivalence theorem
 
31-Mar-2015
Lecture
Adams-Bashforth, Adams-Moulton, BDF methods
Absolute stability
Stability domains (handout)
 
Ch. 5.9
Ch. 7.1-7.2
Ch. 7.3-7.6
31-Mar-2015
Lecture
A-stability
Taylor methods
Runge-Kutta methods (RK Stability Domains)
 
Ch. 8.1-8.3
Ch. 5.6
Ch. 5.7
23-April-2015
Lecture
Upwind, Lax-Friedrichs, Lax-Wendroff methods
Demonstration of various advection schemes
Discussion of final project (MAC Grid)
 
Ch. 10.1-10.4
28-April-2015
Lecture
Solving the Navier-Stokes equations (MAC Grid)
 

Homework problem sets

See "How to Present your Work" for instructions on writing up your homework assignments, and basic mathematical symbols for a description of some symbols that can greatly simplify your mathematical writing.
 
Due date Problem set Notes
29-JAN-2015 HW01 weights.m
17-FEB-2015 HW02 (Updated) trigInterp.m, dst.m, idst.m
27-FEB-2015 HW03 dct.m, idct.m
19-MARCH-2015 HW04 testFdPoisson.m, fd2poisson.m, fd2poissondst.m, fd2poissonmg.m, dst.m, idst.m, dct.m, idct.m
16-APR-2015 HW05  
01-MAY-2015 HW06

Final project

Solve the "Lid-driven cavity" problem at Reynolds number 100 and 1000 and compare to the data from Ghia, Ghia, & Shen 1982.

Your goal is to make various improvements to my "pressure-free projection" code for solving the Navier-Stokes equations. This code can be downloaded here. Your grade will be based on the improvements to the code you make in terms of accuracy and speed. The most important improvement to accuracy will come by changing from upwinding in the advection step to a second-order accurate method. The current bottle-necks in speed occur in the diffusion step and in computing the pressure. Please see the article by Prof. Strang on the discrete cosine transform to see how to use this transform to solve the pressure equation on a cell-centered grid.

Matlab tutorials

  • MATLAB basics.
  • Using scripts.
  • Programming in MATLAB, Part I.
  • Programming in MATLAB, Part II.
  • Formatting output.
  • Miscellaneous


    Please e-mail me regarding any problems with the links on this page.