## 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 __