18.336 Numerical Methods for Partial Differential Equations

 

 

 

Spring 2008

 

Course Instructor: Jean-Christophe Nave ( jcnave@mit.edu  - http://web.mit.edu/jcnave/www/ )

 

Tuesday / Thursday 11:00 – 12:30 in 2-136

 

Office Hours: Wednesday 2-3pm

Course Homepage: http://web.mit.edu/jcnave/www/courses/18.336.htm

 

TA: David Shirokoff (Office: 2-342) [shirokof (at) mit (dot) edu]

TA Office Hours: Tuesday 5-7pm

 

Syllabus

Survey

Some Course notes from P. Koev (‘2005)

 

 

 

Course Description:

 

Advanced introduction to applications and theory of numerical methods for solution of partial differential equations, especially of physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods. We will Start from fundamental solutions of PDE and move toward numerical methods including finite difference methods, spectral methods. Additional topics arising from the discretization and solution of specific equations will be covered, i.e. stability, consistency, convergence, Preconditioned Conjugate Gradient (PCG), Fast Fourier Transforms (FFT), C and Matlab programming.

 

Prerequisites:

 

Some notion of programming using Matlab, C/C++, or Fortran. + courses in Calculus, Linear Algebra, and Differential Equations.

 

Grading: 4 or 5 Homework sets (50%) and final project (Including a final presentation) (50%)

 

Final Projects:

 

I will suggest many final projects’ topics. However, I strongly encourage students to come up with their own topic. The project can be related to their own research, but I will NOT allow recycling or repackaging of previous projects or research. The project has to be original and non-trivial.

 

List of useful references:

 

Spectral Methods in Fluid Dynamics, Canuto, et al.

Computational Techniques for Fluid Dynamics (I and II), C.A.J. Fletcher   

Partial Differential Equations, Lawrence C. Evans

Numerical Recipes

 

 

Final Project Guidelines:

 

The write up should have the format (and professional look) of a journal article. I strongly suggest the format of the journal of computational physics (you should download and use their latex template).

Limit yourself to 20 pages not including bibliographical references, appendices, and code.

 

At least the following section should be included:

 

-1- Background on the problem chosen, relevance with respect to previous work done, and aim of the present work

-2- Governing equations

-3- Discretization / Numerical approach

-4- Stability / accuracy …

-5- Results

-6- Convergence study (grid refinement / time step refinement …)

-7- Conclusion + future work

 

GOOD LUCK!!!

 

Homework:

 

1-    Tentatively due Thursday 2/21/08 or Tuesday 2/26/08: Preliminary project proposal – 1 Page Max.

2-    Due Thursday 2/28/08: Project proposal: clear description of the problem to be solved. This should contain a brief motivation, equations, expected approach (method to be used, programming language), and bibliographical references.

3-      PSET 1 delivered on Tuesday 3/4/08 due Tuesday 3/18/08

4-      Midterm project Report due Thursday 4/10/08

5-      PSET 2 delivered on Thursday 4/10/08 due Thursday 4/24/08

 

 

Lecture 1 – 2/05/08 We settled some administrative issues, went over the syllabus and the survey. I started the formal introduction to PDE’s with some definitions. Hand out 1: Common PDEs

Lecture 2 – 2/07/08 We covered well posedness, and derived solutions for the linear transport equation.

Lecture 3 – 2/12/08 We covered fundamental solutions to the Laplace’s and Poisson’s equations and showed mean-value properties, maximum principle, uniqueness …

Lecture 4 – 2/14/08 We covered fundamental solutions to the heat equation.

Lecture 5 – 2/21/08 Final project discussion

Lecture 6 – 2/26/08 Common Finite Difference approximation for the 1st, 2nd, and higher order derivatives and their analysis (order, efficiency, compactness …)

Lecture 7 – 2/28/08 Simple Boundary Value Problem

Lecture 8 – 3/04/08 Continued on consistency, stability, convergence

Lecture 9 – 3/06/08 Continued on consistency, stability, convergence

Lecture 10 – 3/11/08 Fourier analysis of PDE’s and started diffusion equation

Lecture 11 – 3/13/08 Von Neumann Stability analysis, stiffness of diffusion equation…

Lecture 12 – 3/18/08 Multi-dimensional problem, LOD, ADI…
Lecture 13 – 3/20/08 Elliptic systems discretization, basic solution of the linear system (to be expanded in Lecture 15)

Lecture 14 – 4/01/08 Advection equation, Leap Frog, Lax-Friedrichs…

Lecture 15 – 4/03/08 Guest lecture by Benjamin Seibold on Conjugate Gradient methods and multigrid.

Lecture 16 – 4/08/08 Lax-Wendroff, Beam-Warming, method of characteristics, CFL condition