Math 567 : Finite difference methods for differential equations
This course will introduce you to methods for solving partial differential equations (PDEs) using finite difference methods. The course content is roughly as follows :
- Finite difference approximations of differential operators.
- Local and global truncation error; numerical consistency, stability and convergence; The Fundamental Theorem of Finite Difference Methods.
- Steady state and boundary value problems
- Numerical time stepping methods for ordinary differential equations, including forward Euler, backward Euler, and multi-step and multi-stage (e.g. Runge-Kutta) methods.
- The time dependent heat equation (an example of a parabolic PDE), with particular focus on how to treat the stiffness inherent in parabolic PDEs. Method of lines discretizations. Lax-Equivalence Theorem; Lax-Richtmeyer Stability.
- Numerical methods for PDEs describing wave-like motion (hyperbolic PDEs).
- Numerical methods for mixed equations involving hyperbolic, parabolic and elliptic terms
- Brief introduction (if time permits) to finite volume methods for the discretization of conservation laws derived from physical principles.
- Basic course information
- Required textbook and other resources
- Homework assignments
- Grading policy
Send me an e-mail
Please send me an e-mail at firstname.lastname@example.org so that I can compile an e-mail list for the class. At the very least, include a subject header that says "Math 567". 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|
|Office Hours||Thursday 1:15-2:45|
|Prerequesites||Math 465 (Introduction to Numerical Methods) or equivalent course|
Required textbook and other resources
- Finite Difference Methods for Ordinary and Partial Differential Equations (Time dependent and steady state problems), by R. J. LeVeque. Society for Industrial and Applied Mathematics (SIAM), (2007) (required).
- Finite Volume Methods for Hyperbolic Problems, by R. J. LeVeque. Cambridge University Press, (2002) (suggested).
- A Friendly Introduction to Numerical Analysis, by Brian Bradie. Pearson Prentice Hall, (2006) (suggested).
- Analysis of Numerial Methods, by Eugene Isaacson and Herbert Keller. Dover Books on Mathematics, (1994) (suggested).
- Numerical Computing with Matlab, by Cleve Moler. Mathworks, Inc., (2004) (suggested).
- Learning Matlab, by Toby A. Driscoll. The Society for Industrial and Applied Mathematics, (2009) (suggested).
We will stick the following schedule as much as possible.
|Week #1 (Aug. 22)||
Tuesday -- Introduction to finite difference approximations
Thursday -- Finite difference approximations; Computing truncation errors
|Week #2 (Aug. 28)||
Tuesday -- Chapter 1 : Designing finite difference stencils
Thursday -- Boundary value problems (2.1-2.5)
|Week #3 (Sept. 4)||
Tuesday -- Stability, Consistency and Convergence (2.6-2.10)
Thursday -- Green's Functions and max-norm stability (2.11)
|Week #4 (Sept. 11)||
Tuesday -- Mathworks on-campus visit: River Front Hall, Rm. 101 (9:00-11:00)
Thursday -- Work through details of Section 2.11 on stability of BVP problem
|Week #5 (Sept. 18)||
Tuesday -- Neumann boundary conditions (2.12-2.13)
|Week #6 (Sept. 25)||
Tuesday -- Splitting methods : Jacobi and Gauss Seidel (4.1 - 4.2)
|Week #7 (Oct. 2)||
Tuesday -- Matrix-vector multiply in 2d
Thursday -- Chapter 5 : Initial Value Problems; Basic numerical methods; One step methods
|Week #8 (Oct. 9)||
Tuesday -- Terminology (one-step; multi-step, and so on); Truncation error
Thursday -- Runge-Kutta Methods
|Week #9 (Oct. 16)||
Tuesday -- Midterm #1
Thursday -- Multi-step methods
|Week #10 (Oct. 23)||
Tuesday -- Numerical convergence study of ODE methods
Thursday -- Chapter 6 : Zero stabilty for one-step methods
|Week #11 (Oct. 30)||
Tuesday -- Zero stabilty for linear multi-step methods
Thursday -- Chapter 7 : Absolute stability for ODEs
|Week #12 (Nov. 6)||
Tuesday -- Absolute Stability (cont.)
|Week #13 (Nov. 13)||
Tuesday -- Non-linear problems; Chapter 9 : Diffusion Equation
Thursday -- Diffusion Equation
|Week #14 (Nov. 27)||
Tuesday -- Scalar advection and hyperbolic schemes
Thursday -- Scalar advection and hyperbolic schemes
|Week #15 (Dec. 5)|
Homework assignments are due Thursday, at the start of class.
Due Sept. 7
Due Sept. 22
Due Oct. 5
Due Nov. 10
We will have two midterms and (possibly) a final exam
|Midterm #1||Date: TBA|
|Midterm #2||Date: TBA|
You can find the Final Exam calendar here.
Homework will count for 60% of your final grade, and the rest will be made up with exams and a possible final or final project.