M 383G / CAM 386K : Numerical Treatment of Differential Equations

Fall 2006 (60350 / 66475)

General Information

Instructor: Oscar Gonzalez (og@math.utexas.edu) 471-1138 Office hrs: RLM 10.122, Tu 10-11am, W 2-3pm, Th 8-9am Location: RLM 5.120, MWF 1-2pm Text: (none required) Web: http://www.ma.utexas.edu/~og/TEACHING/M383G/m383g.html Links: schedule, references

Description

This course is an introduction to the approximation of differential equations by numerical methods. It is aimed primarily at first-year graduate students and is designed to give students practical experience in the analysis and implementation of methods for various types of initial and boundary value problems arising in the engineering and physical sciences. The main methods to be considered are one-step and multi-step methods for systems of ordinary differential equations, and finite difference and finite element methods for elliptic, parabolic and hyperbolic partial differential equations. The goals are to analyze the convergence and stability properties of various classic methods applied to standard model problems, carry out numerical experiments which illustrate various properties of the methods, and discuss generalizations.

Prerequisites

Students will be expected to have a working knowledge of linear algebra, multivariable calculus, differential equations, and elementary numerical analysis. It is also expected that students will have some basic programming skills and some familiarity with mathematical software such as Matlab. Efforts will be made to accommodate students with different backgrounds.

Policies

Homeworks: Homework will be assigned throughout the semester (about every two weeks or so) and will contain a mix of analytical and computational problems.


Matlab: Various homework problems in the course will require the use of Matlab, which is an easy-to-use software package available in many computer systems around campus. Prior experience with Matlab is not required and various primers and references are available:

• Matlab primer (based on Matlab 3) postscript document
  
• Matlab primer (based on Matlab 4) postscript document
  
• Matlab online reference documentation (based on Matlab 4) html document
  
• A practical introduction to Matlab (based on Matlab 5) postscript document
  

Exams: There will be no exams.

Course grade: Course grades will be based entirely on the homework sets. Students who do not want to do all the homeworks should register on a Credit/No-Credit basis.

Tentative schedule

The following schedule will likely change as the semester progresses and the actual material to be covered will depend on student interest and time.


Part I. Systems of ordinary differential equations

• One-step, Multi-step and Runge-Kutta methods
  
• Truncation error, order, convergence
  
• Long-time behavior, A-stability
  
• Implementation issues, Newton's method
  

Part II. Elliptic partial differential equations

• Strong formulation, Finite difference methods
  
• Maximum principle, convergence
  
• Variable coefficients, curved boundaries
  
• Weak formulation, Finite element methods
  
• Best approximation property, error estimates
  

Part III. Parabolic partial differential equations
• Euler, Crank-Nicholson finite difference methods
  
• Fourier stability, Maximum principles
  
• Truncation error, order, convergence
  
• Variable coefficients, nonlinearities
  

Part IV. Hyperbolic partial differential equations
• Upwind and Lax-Wendroff finite difference methods
  
• Characteristics, Domain of dependence, CFL condition
  
• Fourier stability, Phase and amplitude errors
  

Helpful references

• Numerical analysis. An introduction, W. Gautschi, Birkhauser, 1997.
• Numerical methods for ordinary differential systems, J.D. Lambert, Wiley, 1991.
• The numerical analysis of ordinary differential equations, J.C. Butcher, Wiley, 1987.
• Numerical methods for two-point boundary-value problems, H.B. Keller, Dover, 1992.
• Numerical solution of partial differential equations, K.W. Morton & D.F. Mayers, Cambridge, 1994.
• Numerical solution of partial differential equations by the finite element method, C. Johnson, Cambridge, 1987.
• Numerical methods for conservation laws, R. J. LeVeque, Birkhauser, 1992.