Math 517 – Finite Difference Method

Course Information   

 Instructor:

 Yan, Jue

 Course room:

 Carver Hall 0032

 Office Hours:

 Tu & Th 2:00-3:10 PM (or by appointment)

 E-mail:

 jyan@iastate.edu

 Office Number:

 294-8166 (Carver 484)

 

Textbook: Time Dependent Problems And Difference Methods, by B. Gustafsson, H-O Kreiss and J Oliger.

 

Homeworks:

               

HW#

Out

Due

#1: 1.1.2 and 1.2.1

Aug 23

Aug 30

#2: 2.1.3, 2.3.1 and 2.4.2

Sep 6

Sep 13

#3: 2.5.2, 2.5.3 and 2.7.1

Sep 19

Oct 2

#4: 2.6.1, 2.8.1,4.2.1 and 4.3.1

Oct 9

Oct 18

      Midterm Exam

 

 

#5: 5.1.3, 5.2.2 and 5.3.3

Nov 15

Nov 29

    Final Exam

Nov 30

Dec 14

 

 

Outline of Lectures

 

Chapter 1 – Review on Fourier analysis for smooth and non-smooth functions. 

Chapter 2 – Using 1D wave equation to introduce stability and consistency of a finite difference numerical scheme.

Lax-Friendrichs method, Lax-Wendrrof method and Leap-Frog methods as explicit methods are discussed in details.

                                Crank-Nicholson method as an implicit scheme is studied.

Similar schemes for heat equation and convection-diffusion equation are introduced.

                Chapter 3 – General idea for higher order difference scheme is introduced for the model equations

                Chapter 4 – Establish well-posed theory for general PDEs.

                                1D scalar hyperbolic and parabolic equations.

                                1D linear hyperbolic and parabolic systems with constant coefficients.

                                General systems with constant coefficients.

                Chapter 5 – General stability and convergence analysis are established with hyperbolic system as a model problem.  

Parallel to PDE well-posedness theory, numerical stability is introduced for difference scheme.

                                Local truncation error and consistency are revisited.

                                Lax convergence theorem. Stability + consistency = convergence.

                                von-Neumaan condition. Dissipation. Splitting method. ADI method.

                Chapter 6 – Classification for hyperbolic systems, strictly, symmetric, strongly and weakly hyperbolic.

                                Revisit well-posedness theory for constant coefficient and variable coefficient problems.

                                Extension of well-posedness study to multi-dimensional hyperbolic systems.

                                General guideline of numerical methods for hyperbolic systems.

                                Method of lines(time and space discretization seperately). Runge-Kutta method.

                                Introduction on finite volume method for conservation laws.

                Chapter 7 – General parabolic systems. Guideline for numerical methods.

 

Interesting future topics

 

1)       Numerical methods for linear wave equation with discontinuous solutions.

2)       Conservative numerical methods for nonlinear conservation laws.

3)       Higher order numerical methods (TVD, ENO, WENO).

 

References

1)       Numerical Methods for Conservation Laws, by Randall J. LeVeque.

2)      Numerical Partial Differential Equations – Finite Difference Methods, by J. W. Thomas.

 

Grading Policy

Course grades will be determined from student performance on homework assignments. There will be six to eight homework assignments, one midterm and one take home final.

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Last modified: Friday, November 30, 2007