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.
Discussion Section: Thursday, 1.00PM - 1.50PM, MS 5117
Teaching Assistant: Jean-Michel Maldague
Office: MS 2963
Office hours: Thursday 2-4pm.
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).
- 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
Course Requisites: Math 115A, Math 151A, and Math 151B or equivalent.
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.1.18f/
Numerical Analysis Qualifying Exam
Numerical Recipes in C
Getting started with MATLAB
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)
HW #2 (due on Monday, October 15)
HW #3 (due on Friday, October 26)
HW #4 (due on Monday, November 5)
Sample practice problems for the midterm
HW #5 (due on Friday, November 16)
HW #6 (due on Wednesday, November 28, or on Friday, November 30)
HW #7 (due on Friday, December 7)
Sample practice problems for the final and announcements