## Math 269A, Section 1, Fall 2018

## Advanced Numerical Analysis

### Numerical solution for systems of ordinary differential equations; initial and boundary value problems

**Lecture Meeting Time:** MWF 1.00PM - 1:50PM.

** Lecture Location:** MS 5138

**Instructor:** Luminita A. Vese

**Office:** MS 7620-D

**Office hours:** Wednesday 2-2.45pm, or by appointment.

**E-mail:** lveseATmath.ucla.edu

**Discussion Section:** Thursday, 1.00PM - 1.50PM, MS 5117

**Teaching Assistant:** Jean-Michel Maldague

**Office:** MS 2963

**Office hours:** Thursday 2-4pm.

** E-mail:** jmmaldagueATmath.ucla.edu

**Recommended Textbooks:**
U.M. Ascher and L.R. Petzold,
*"Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations"*, SIAM 1998.
A. Iserles, "*A first course in the Numerical
Analysis of Differential Equations"*, Cambridge Texts in Applied
Mathematics, Cambridge University Press, 1998.
J. Stoer and R. Bulirsch, *"Introduction to Numerical Analysis"*, Third Edition, Springer 2002. (Chapter 7).

**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

**Course Requisites:** Math 115A, Math 151A, and Math 151B or equivalent.

**Useful Links:**
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.1.18f/
Numerical Analysis Qualifying Exam
Numerical Recipes in C
Getting started with MATLAB

**Homework Policy:**
There will be several assignments involving both theoretical and computational exercises. These will be collected on Friday in class.

**Examinations:** A one-hour final exam.

__Final Exam:__ Friday, December 7, 1-2pm. There will be a one-hour final exam during the last lecture.

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 70%, Final 30%

**Weekly Homework Assignments:**

**HW #1** (due on Friday, October 5)

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HW1.pdf__

**HW #2** (due on Monday, October 15)

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HW2.pdf__

**HW #3** (due on Friday, October 26)

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HW3.pdf__
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HW3.tex__

**HW #4** (due on Monday, November 5)

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HW4.pdf__
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HW4.pdf__

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Sample practice problems for the midterm __

**HW #5** (due on Friday, November 16)

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HW5.pdf __
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HW5.tex __

**HW #6** (due on Wednesday, November 28, or on Friday, November 30)

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HW6.pdf __
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HW6.tex __

**HW #7** (due on Friday, December 7)

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HW7.pdf __
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HW7.tex __

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Sample practice problems for the final and announcements __