AMATH 741/CS 778

Numerical Solution of Partial Differential Equations

Is offered every winter term.

Brief description:

Many problems in science, engineering, finance and economics can be formulated in terms of Partial Differential Equations (PDEs). Since analytical solutions are not available in general, it is necessary to use numerical methods to approach the solution. This course will cover the basic techniques for solving PDEs numerically. The goal of the course is threefold. You will receive a solid introduction to the theory of numerical methods for partial differential equations (with derivations of the methods and some proofs). You will learn to implement the computational methods efficiently in Matlab, and you will apply the methods to problems in several fields, for example, fluid mechanics, diffusion processes and wave phenomena. You will also apply the course material to a problem in your own field of research in a project.

Intended audience:

This course is a core course for graduate students in Applied Mathematics. It will also be of interest to students in Computer Science (Scientific Computing), Science (Physics, Chemistry, Biology, ...), Engineering (especially Mechanical Engineering) and Computational Finance.

Prerequisites:

Some previous experience with numerical computation (e.g. CM 271/AMATH 341/CS 370 or CS 371) and MATLAB, and some background in PDEs is desirable.

Outline:

Classification of PDEs
Finite Difference methods
Finite Volume methods for conservation laws
Finite Element methods
Implicit schemes and iterative solvers
Spectral methods/

References:

"Numerical Partial Differential Equations: Finite Difference Methods", J.W. Thomas, Springer, 1995.
"Finite Volume Methods for Hyperbolic Problems", by Randall J. LeVeque, Cambridge University Press, 2002.
"Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics", by Dietrich Braess, Cambridge University Press, 2001.
"An Introduction to the Finite Element Method", by J. N. Reddy, McGraw-Hill, 1993.