Math 269A, Section 1, Fall 2002
Advanced Numerical Methods
Lecture Meeting Time: MWF 2.00PM - 2:50PM.
Lecture Location: MS 5148.
Instructor: Luminita A. Vese
Office: MS 7620-D
Office hours: MW 3-4PM, F 1-2, or by appointment (subject to change).
Discussion Section: Thursday, 3.00PM - 3.50PM, MS 5127
Teaching Assistant: Jorge Balbas
Office: MS 2529.
Office hours: W 1.00pm - 2.00pm (MS 2529), and F 3-4 (PIC lab BH 2817).
J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis", Second
Edition, Springer 1992. (Chapter 7).
Recommended Textbook: A. Iserles, "A first course in the Numerical
Analysis of Differential Equations", Cambridge Texts in Applied
Mathematics, Cambridge University Press, 1998.
- Notations and terminology for ODE's and systems of
ODE's; reduction of higher order ODE's to 1st order systems of ODE's;
the fundamental existence and uniqueness thm. for ODE's (Lipschitz
- Introduction of Euler's method, order of Euler's method,
methods (introduction, definition, consistency, local truncation error).
- Explicit Runge-Kutta (ERK) methods (introduction of the method
in the general case, notations in the general case, derivation of ERK of
second order); Runge-Kutta method of fourth order.
- Examples of implicit methods: trapezoidal rule,
midpoint rule, the theta method, and the implicit Euler's method; computation
of orders for these methods.
- Convergence of one-step methods (the general case; see also convergence for
Euler's method, etc).
- Asymptotic expansions for the global discretization error for one step
methods, and applications to error estimate.
- Practical implementation of one step methods
- Linear Multistep methods: examples, derivation using the Lagrange
- Linear multistep methods: definition and computations of the local
truncation error, order of the method, consistency.
- Implicit and explicit linear multistep methods; predictor-corrector methods.
- Examples of consistent multistep methods which diverge.
- Linear difference equations: stability (root) condition, general solution.
- Convergence Thm. for linear multistep methods
- Order and consistency for linear multistep methods
- Adaptive methods for one-step and multi-step methods, error control,
Milne device, extrapolation
- Stiff differential equations, stability and intervals (regions) of absolute
stability, A-stable methods, BDF methods
- Numerical methods and stability for systems of ODE's
- Finite difference methods for linear BVP
- Functional (fixed point) iteration and Newton's iteration for solving
systems of ODE's using an implicit method
Prerequisites: Math 115A, Math 135A, Math 151A, Math 151B or equivalent.
Virtual Office Hours
PIC Lab: Boelter Hall 2817 and
Mathematical Sciences 3970
MATLAB Documentation (thanks to Prof. C. Anderson, UCLA)
Class Web Page: http://www.math.ucla.edu/~lvese/269a.1.02f/
Numerical Analysis Qualifying Exam
Numerical Recipes in C
Getting started with MATLAB
Weekly assignments involving both theoretical and computational exercises.
Examinations: One midterm exam and one final exam.
Midterm Exam: Friday, November 1st, (lecture time and location).
Final Examination Code: 07 - Monday, December 9, 11.30AM - 2.30PM.
The examinations are closed-book and closed-note.
No exams at a time other than the designated ones will be allowed
(exceptions for illness with document proof, or emergency).
Grading Policy: HW 40%, Midterm 20%, Final 40%
Weekly Homework Assignments:
HW #1 (due Monday, October 7)
Matlab Code for Euler's Method
C++ Code for Euler's Method
HW #2 (due Friday, October 11)
Matlab Code for Fourth-Order Runge-Kutta Method
C++ Code for Fourth-Order Runge-Kutta Method
HW #3 (due Friday, October 18)
HW #4 (due Friday, October 25)
Sample Matlab Code for Adams-fourth order predictor-corrector algorithm
Sample C++ Code for Adams-fourth order predictor-corrector algorithm
Sample practice problems for the midterm
HW #5 (due Wednesday, November 13)
HW #6 (due Friday, November 22)
HW #7 (due Thursday, December 5)
Sample practice problems for the final and announcements