MATH 578: Numerical Partial Differential Equations

Time and Place: Tuesday/Thursday 11:00 am - 12:15 pm, MITCH 121
Instructor: Jehanzeb H. Chaudhry (Zeb),,
Office Hours: Tuesday: 1:30 - 3:00 pm, Wednesday: 9:00 - 10:30 am (SMLC 328)

Texts Recommended (not required):

* Notes on all methods

* Finite Difference Methods

* Finite Volume Methods

* Finite Element Methods

Prerequisites: Math 463/513, Math 504, Math 505

Course Description:   This course covers the basics of finite difference schemes, finite volume schemes, and finite element methods. Additional topics (e.g. a posteriori error estimation, discontinuous Galerkin methods, etc) may be covered based on student interest and time constraints.You do not need to be an expert in PDEs or in coding. But you should have a course in numerical analysis as your background, be comfortable with differential equations, and have some coding experience. We'll be covering numerical methods for parabolic, hyperbolic and elliptic equations. We'll discuss the mathematical background of the numerical methods as well as its implementation.



I'll be using Python for the examples in class. The course homeworks and examples in class will be in Python with the libraries numpy, scipy and matplotlib for assignments. Python has a very gentle learning curve, so you should feel at home even if you've never done any work in Python. While it is possible to do the finite difference HW's in MATLAB, I highly recommend you do these in Python, since Python will be needed once we move to the Finite Element package Dolfin from the FEniCS Project.

*Python and Numpy Help

*Virtual Machine

Virtual Box Image Username: mathgeeks Password: abc123

45% Homework
45% Projects
10% Participation

Homework 0
Homework 1 Due: September 13
Update 1: Sept 2nd. Added domain for Problem 1. Added a bonus problem. Solution Code
Homework 2 Due: September 27 September 29
Update 1: Sept 20th. Fixed missing negative sign on RHS in Problem 5.
Update 2: Sept 26th. Due Date changed to Sept 29.
Homework 3 Due: October 11
Update 1: Oct 6th. Fixed sign issue in problem 3, equation numbered 2. hw3_fd hw3_fv
Homework 4 Due: November 3
Homework 5 Due: November 24   elements.dat  vertices.dat  dom.xml  dom_physical_region.xml

Midtem Project Due: November 8. You can work in groups of 2.  Addendum
Final Project Due: December 13. You can work in groups of 2. 


Sample files/ Class Examples

numpy_matplotlib_example   explicit scheme for heat equation  explicit scheme for heat equation with mat-vec operations  Method of Lines Example 
1D FEM  1D FEM in Dolfin  1D FEM mesh (needed for Dolfin example) 

Handwritted and probably illegible notes: 1-3  4-6  FEM-1  FEM-2 

Final Project Presentations: December 13

Time Title Presenter(s)
12:30 Stokes Flow through a Backward Facing Step Robert Malakhov
12:45 Adjoint Based Error Estimation and Mesh Refinement Juan Diego Colmenares Fernandez
1:00 Using proper orthogonal decomposition to obtain solution from a parametrized partial differential equation Joel Upston
1:15 Comparison of Least Squares and Mixed Method for Stationary 2D Convection-Diffusion Equation Shu Wang and Anastassiya Semenova
1:30 Clement's Interpolation Adam Frederickson
1:45 Hermite interpolation method on the overset grids Oleksii Beznosov
2:00 Petro-Galerkin 3-Dimensional Burgers Equation Brad Philipbar