##
18.336: Numerical Methods of Applied Mathematics -- II, Spring 2005

** Where and when:** 2-151, MW 3-4:30

** Introduction: **
Advanced introduction to applications and theory of numerical methods for
solution of differential equations, especially of physically-arising
partial differential equations, with emphasis on the fundamental ideas
underlying various methods. Topics include finite differences, spectral
methods, well-posedness and stability, boundary and nonlinear instabilities.
The course assumes familiarity with basic (numerical) linear algebra and
will involve some programming.

** Textbooks (optional): **

Spectral Methods, Trefethen, SIAM 2000

Finite Difference
Schemes and Partial Differential Equations, John Strikwerda

** Instructor:** Plamen Koev, office: 2-376,
phone: 253-5013, e-mail: plamen (at) math (dot) mit (dot) edu
**
**

Office hours: Monday 4:30-6:30

** Grading: ** One midterm on March 16, 2005 and
about six homework assignments two of which may be projects. You may
cooperate
on the problem sets, but you must write up and turn in your own
solutions, and be able to explain them.

** Lectures:** Lecture Notes (updated
continuously throughout the semester).

02/02 Introduction, Examples

02/07 Wave equation--Consistency, Stability, Convergence

02/09 Von Neumann analysis for stability

02/14 Von Neumann analysis for stability - II

02/16 Leap-frog

02/22 Dissipation Numerical Example

02/23 Dispersion

02/28 Dispersion II

03/02 Group Velocity and Propagation of Wave Packets I.
Numerical Example

03/07 Group Velocity and Propagation of Wave Packets II

03/09 Parabolic Equations

03/14 The Du Fort Frankel Scheme.
Numerical Example

03/16 Midterm.
Practice Midterm.

03/28 The Convection-Diffusion equation I
Numerical example

03/30 The Convection-Diffusion equation II

04/04 Systems of differential equations

04/06 The solution of the system u_{t}+Au_{x} = Bu

04/11 The solution of the system u_{t}+Au_{x} = Bu_{y}

04/13 ADI Methods

04/20 Elliptic Equations

04/25 Numerical methods for Elliptic equations

04/27 Detailed analysis of Jacobi, Gauss-Seidel and SOR(w)

05/02 Multigrid, part I (Example mgdemo.m and others located in here).

05/04 Multigrid, part II

05/09 Numerical methods for ODEs.
Handout.
Numerical Example.

05/11 Stiff ODEs

Software used for demonstration in class.

** Homework: **

- Due 2/23/05. Prove that the Crank-Nicolson scheme (notes, p.9,
bottom) is unconditionally stable.
- Due 3/2/05. PDF.
- Due 3/9/05. PDF.
- Due 4/13/05. PDF.
- Due 4/27/05. PDF.
- Due 5/11/05. PDF.