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).
E-mail: lvese@math.ucla.edu

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).
E-mail: jbalbas@math.ucla.edu

Required Textbook: 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.

Topics:
- 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 condition).
- Introduction of Euler's method, order of Euler's method, one step 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 interpolation polynomial
- 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.

Useful Links:
  • Virtual Office Hours
  • PIC Lab: Boelter Hall 2817 and Mathematical Sciences 3970
    http://www.pic.ucla.edu/piclab/
  • 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
    file:/net/fern/m4/matlab-5.3.11/help/techdoc/basics/getstarted.html

    Homework Policy: 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)
    HW1.pdf
    Matlab Code for Euler's Method
    C++ Code for Euler's Method

    HW #2 (due Friday, October 11)
    HW2.pdf
    Matlab Code for Fourth-Order Runge-Kutta Method
    C++ Code for Fourth-Order Runge-Kutta Method

    HW #3 (due Friday, October 18)
    HW3.pdf

    HW #4 (due Friday, October 25)
    HW4.pdf
    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)
    HW5.pdf

    HW #6 (due Friday, November 22)
    HW6.pdf

    HW #7 (due Thursday, December 5)
    HW7.pdf

    Sample practice problems for the final and announcements