Math 621 - Numerical Methods for Partial Differential Equations

Spring 2015 - Matthias K. Gobbert

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Presentations of the Class Projects

The presentations of the class projects will be on Tuesday, May 19, 2015, 01:00 p.m. in SOND 109. Please follow the link to the Program for the titles and abstracts. Just like for seminar talks, everybody is welcome to attend!

Basic Information

Course Description

Many models for physical processes in nature and in engineering consist of partial differential equations. The models are as varied as reality itself, but often non-linear and often involving systems of partial differential equations. In all but some textbook examples, an analytic solution is impossible. That necessitates the use of numerical methods for partial differential equations, and this area forms a vast field itself and is one of the major driving forces behind research in many other fields like numerical linear algebra, scientific computing, and the development of parallel computers.

Despite their many forms, many equations share certain fundamental mathematical properties and can be classified into the three basic categories of elliptic, parabolic, and hyperbolic partial differential equations. It makes therefore sense to study the mathematical properties and numerical methods for prototype equations of each type. Classical examples for the three types are the Poisson equation, the heat equation, and the scalar transport equation, respectively. This course will provide an overview of the types of equations, their most fundamental mathematical properties, and demonstrate numerical methods for them. It will have a strong focus on parabolic problems such as systems of time-dependent reaction-diffusion equations, but we will also cover elliptic and hyperbolic prototype problems.

Two large classes of methods are finite difference and finite element methods, and we will discuss examples of both methods. We will use this as the basis for discussing the associated issues of discretizing the time-direction and solving large sparse systems of linear equations efficiently with respect to memory and computing time. One specific goal of this course is to understand the method of lines approach to transient reaction-diffusion equations including all numerical techniques necessary to deal with the spatial and time discretizations as well as non-linear and linear solvers.

For the finite difference methods, we will write our own code; we will use MATLAB for this purpose because of its ease of programming. You should have a foundation in using Matlab, equivalent to its Getting Started guide; see a link to my Matlab webpage below. But you should expect to learn additional commands and techniques to get the best resolution and fastest performance. For the finite element method, the course uses the state-of-the-art finite element package COMSOL Multiphysics to give you experience with a full-featured professional-grade numerical package.

We will focus both on computational experiments and on rigorous mathematical analysis of the numerical methods considered, but with a slant towards the computational side of the subject. This course was originally designed for second-year graduate students in Applied Mathematics, hence you are ordinarily expected to have knowledge of Numerical Analysis (Math 620) and Numerical Linear Algebra (Math 630), though we will cover all crucial concepts from the background briefly. The course is taught without expecting a formal background in the theory of partial differential equations, but you should be ready to learn some background information when necessary. The approach of this course is designed to accomodate a varied background of the audience, and it has been successfully taken both by applied mathematics students with different ranges of experience and by scientists and engineers from a variety of departments. Please contact me if you have any questions about the expectations.

Learning Goals

By the end of this course, you should:

Other Information

UMBC Academic Integrity Policy

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. To read the full Student Academic Conduct Policy, consult the UMBC Student Handbook, the Faculty Handbook, the UMBC Integrity webpage, the UMBC Undergraduate Student Academic Conduct Policy (PDF) for undergraduate students, or the University of Maryland Graduate School, Baltimore (UMGSB) Policy and Procedures for Student Academic Misconduct (PDF) for graduate students.

Copyright © 2001-2015 by Matthias K. Gobbert. All Rights Reserved.
This page version 2.2, May 2015.