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 multistep and multistage (e.g. RungeKutta) 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. LaxEquivalence Theorem; LaxRichtmeyer Stability.
 Numerical methods for PDEs describing wavelike 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
 Lectures
 Homework assignments
 Exams
 Grading policy
Send me an email
Please send me an email at donnacalhoun@boisestate.edu so that I can compile an email 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  Mathematics 241A 
Time  Tuesday/Thursday 9:0010:15 
Place  Math 124 
Office Hours  Thursday 1:152:45 
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).
Lectures
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 (2.12.5)

Week #3 (Sept. 4) 
Tuesday 
Stability, Consistency and Convergence (2.62.10)
Thursday 
Green's Functions and maxnorm stability (2.11)

Week #4 (Sept. 11) 
Tuesday 
Mathworks oncampus visit: River Front Hall, Rm. 101 (9:0011:00)
Thursday 
Work through details of Section 2.11 on stability of BVP problem

Week #5 (Sept. 18) 
Tuesday 
Neumann boundary conditions (2.122.13)
Thursday 
Variable coefficient problems (2.15); Direct methods for
2d elliptic problems (3.13.3)

Week #6 (Sept. 25) 
Tuesday 
Splitting methods : Jacobi and Gauss Seidel (4.1  4.2)
Thursday 
Steepest descent methods; Conjugate Gradient (4.3)

Week #7 (Oct. 2) 
Thursday 
Chapter 5 : Initial Value Problems;
Basic numerical methods; One step methods

Week #8 (Oct. 9) 
Tuesday 
Terminology (onestep; multistep, and so on); Truncation error
Thursday 
RungeKutta Methods

Week #9 (Oct. 16) 
Tuesday 
Midterm #1
Thursday 
Multistep methods

Week #10 (Oct. 23) 
Tuesday 
Numerical convergence study of ODE methods
Thursday 
Chapter 6 : Zero stabilty for onestep methods

Week #11 (Oct. 30) 
Tuesday 
Zero stabilty for linear multistep methods
Thursday 
Chapter 7 : Absolute stability for ODEs

Week #12 (Nov. 6) 
Tuesday 
Absolute Stability (cont.)
Thursday 
Plotting of stability regions

Week #13 (Nov. 13) 
Tuesday 
Nonlinear problems; Chapter 9 : Diffusion Equation

Week #14 (Nov. 27) 
Tuesday 
Scalar advection and hyperbolic schemes
Thursday 
Scalar advection and hyperbolic schemes

Week #15 (Dec. 5) 
Homework assignments
Homework assignments are due Thursday, at the start of class.
Homework #1 
Due Sept. 7

Homework #2 
Due Sept. 22

Homework #3 
Due Oct. 5

Homework #4 
Due Nov. 10

Exams
We will have two midterms and (possibly) a final exam
Midterm #1  Date: TBA 
Midterm #2  Date: TBA 
Final  Date: TBA

You can find the Final Exam calendar here.
Grading policy
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.