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 ut+Aux = Bu
04/11 The solution of the system ut+Aux = Buy
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:
  1. Due 2/23/05. Prove that the Crank-Nicolson scheme (notes, p.9, bottom) is unconditionally stable.
  2. Due 3/2/05. PDF.
  3. Due 3/9/05. PDF.
  4. Due 4/13/05. PDF.
  5. Due 4/27/05. PDF.
  6. Due 5/11/05. PDF.