MA8404 Numerical solution of time dependent differential equations, fall 2015
Lectures
- Room 822
- Tuesdays 10:15-13:00, with a 30 minute lunch break at 11:45
- Wednesdays 12:30-14:00
Meetings during the week of 21-25 September are:
- 16:00-17:00 on Monday, 21 September, we meet in 734
- 12:00-13:00 on Wednesday, 23 September, we meet in 734
Instructor
- Brian E. Moore, Guest Associate Professor from the University of Central Florida
- E-mail: brian.moore@ucf.edu
- Office: 1348
Resources for Learning
- Hairer, Lubich, Wanner, Geometric Numerical Integration (GNI), Springer (2000)
- Leimkuhler and Reich, Simulating Hamiltonian Dynamics (SHD), Cambridge (2004)
- Harier, Norsett, Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (SODE I) 2nd ed, Springer (1993)
- Harier, Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations (SODE II) 2nd ed, Springer (1996)
- Thomas, Numerical Partial Differential Equations: Finite Difference Methods (NPDE), Springer (1995)
Course Goals
The aim of this course is to teach students how to derive, analyze and implement numerical methods for solving time dependent differential equations (ODEs and PDEs). To achieve these aims, students will numerically solve mathematical problems and mathematically analyze the methods used for numerical solution. This course will cover the basic topics of stability, accuaracy, and efficiency for various numerical algorithms. By the end of the semester students should be able to
- Select and implement numerical methods that are most suitable for a given application.
- Determine the stability, order of accuracy, and structure-preserving qualities of a given numerical method.
- Participate in scientific discussions and simple research related to the course topics.
Tentative Schedule
| Week | Topics | Relevant Texts | Suggested Exercises |
|---|---|---|---|
| 17.08 - 21.08 | Information meeting, Room 734, 10:00-11:00, 18 August | ||
| 24.08 - 28.08 | Preliminaries, Euler's Method, Runge-Kutta Methods | ||
| 31.08 - 04.09 | Collocation, Composition and Splitting Methods | GNI II; IV.1-2; SHD 2 | GNI II.6 #2,4,5,6(a); SHD 2.7 #2,8 |
| 07.09 - 11.09 | Numerical Preservation of Invariants, Symmetry, and Symplecticity | GNI V.1-3; VI.1-4; SHD 3,4 | GNI VI.8 #1,4; SHD 3.7 #3,6; 4.6 # 2,11 |
| 14.09 - 18.09 | No Lectures | ||
| 21.09 - 25.09 | Individual Meetings to Discuss Projects, Class meetings:16:00-17:00, 21 Sep. in 734; 12:00-13:00, 23 Sep. in 734 | SHD 2.7 #6; 4.6 #5(b) | |
| 28.09 - 02.10 | Backward Error Analysis | GNI IX.1-4; SHD 5 | GNI IX.11 #5; SHD 5.5 #1,8; 12.4 #6 |
| 05.10 - 09.10 | No Lectures | ||
| 12.10 - 16.10 | Multi-Symplectic Integration | SHD 12 | SHD 12.4 #1,2,4, |
| 19.10 - 23.10 | Equations with Dissipation, Exponential Integrators | ||
| 26.10 - 30.10 | Multi-Step Methods | SODE I: III.1-3 | SODE I: III.2 #1(a),2,5 |
| 02.11 - 06.11 | Stiff Equations | SODE II: IV | SODE II: IV.2 #5; IV.3 #7; IV.7 #3, IV.9 #3 |
| 09.11 - 13.11 | Highly Oscillatory Equations and Parabolic PDEs | GNI XIII.1-4; NPDE 4 | GNI XIII.9 # 1, 2; NPDE 4.3.1, 4.4.6 |
| 16.11 - 20.11 | Problem solving sessions and project presentations | ||
| 14 December | Final Written Exam: 13:00-16:00 in room 922 |
Project
Choose a topic related to your own Master or PhD studies and use the methods/techniques developed in this course in conjunction with your own research. Projects must be subitted in written form, no longer than 10 pages, not including supplementary material such as a cover page or bibliography. Projects must also be presented orally to the class at the end of the semester.
Exam
There will be a written final exam. Details will be posted as they become available.