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Prerequisites

Calculus (18.02), Differential Equations (18.03) or Honors Differential Equations (18.034).

Textbooks

This course as taught during the Spring 2006 term on the MIT campus used the following text:

Strang, Gilbert. Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press, 1986. ISBN: 9780961408800. (Table of Contents)

Since that time, Professor Strang has published a new textbook that is being used for this course as it is currently taught on the MIT campus, as well as for Mathematical Methods for Engineers I (18.085). Information about the new book can be found at the Wellesley-Cambridge Press Web site, along with a link to Prof. Strang's new "Computational Science and Engineering" Web page developed as a resource for everyone learning and doing Computational Science and Engineering.

 Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.

Description

This course has two major topics:

1. Initial Value Problems
   • Linear: Wave Equation, Heat Equation, Convection Equation
   • Nonlinear: Conservation Laws, Navier-Stokes Equation
   • Finite Difference Methods: Accuracy and Stability
   • Lax Equivalence Theorem: CFL and Von Neumann Conditions
   • Fourier Analysis: Diffusion, Dissipation, Dispersion
   • Separation of Variables and Spectral Methods
2. Solution of Large Linear Systems
   • Finite Differences, Finite Elements, Optimization
   • Direct Methods: Reordering by Minimum Degree
   • Iterative Methods and Preconditioning
     • Simple Iteration (Jacobi, Gauss-Seidel, Incomplete LU)
     • Krylov Methods: Arnoldi Orthogonalization
     • Conjugate Gradients and GMRES
     • Multigrid Methods
   • Inverse Problems and Regularization

Requirements

There are no exams in 18.086. Two computational projects take their place, one on each of the major topics in the course. The projects are chosen by each student and they include a brief report.