Numerical Simulation-ME210B (cross-listed CS/ECE/ChemE/Math)
Winter 2012
Professor:
Linda R. Petzold
Departments of Mechanical Engineering and Computer Science,
UCSB
Phone: (805)893-5362 (office), (805)893-5435 (FAX)
Email: petzold@engineering.ucsb.edu
Office: Harold Frank Hall 5107
Office Hours: Fridays 1am-3pm, or email for an appointment
TA:
Michael Lawson
BMSE program
Email: mjl@cs.ucsb.edu
Office Hours: Tuesdays 1pm-3pm in Harold Frank Hall 5106, or email for an appointment
Course Description:
Development of modern numerical methods for ordinary differential equations including Runge-Kutta and multistep.
Convergence, order and stability analysis. Sensitivity analysis. Concepts and capabilities of mathematical software.
Expected Outcomes:
- Understanding of fundamental concepts of numerical solution of ODEs, including
sources and propagation of error, and stiffness.
- Experience with practical issues of software development including error control and stepsize selection.
- Capability of applying stability and accuracy concepts from numerical ODEs to numerical PDEs.
Course
materials:
Class Calendar:
- January 9 - Start Homework 1
- January 16 - UCSB holiday
- January 18 - Homework 1 due, Start Homework 2
- January 25 - Homework 2 due, Start Homework 3
- February 1 - Part 1 of Homework 3 due
- February 6 - Exam 1
- February 8 - Part 2 of Homework 3 due
- February 15 - Start Homework 4 due
- February 20 - UCSB holiday
- February 27 - Part 1 Homework 4 due, Start Homework 5
- February 29 - Homework 5 due, Start Homework 6
- March 7 - Part 2 of Homework 4 due
- March 12 - Homework 6 due
- March 14 - Exam 2
Announcements:
- HW3: Questions 1-3 are due on February 1, Questions 4-5 are due on Febrauay 8
- Professor Petzold will be out of town February 17 and February 24, so there will be no office hours on those days.
- For the computing homeworks, turn in your CODES as well as your results!
- Homeworks may be turned in early to the homework box
- February 8 covered a variable-stepsize derivation of Adams and BDF methods that is
more general than the constant-stepsize derivation in the book. You are responsible for
the variable-stepsize derivation. Here are the notes on that: Notes
Handouts and Other Information:
Lecture Topics and Approximate Schedule (subject to change):
- 1: Standard Form, Existence/Uniqueness Theorem (Section 1.1), Method of Lines (Scholarpedia article), Problem Stability (Section 2.1)
- 2: Stability for Linear Constant-Coefficient and Nonlinear systems (Sections 2.2 and 2.4), Forward Euler Method (Section 3.1)
- 3: Convergence, Accuracy, Consistency (Section 3.2)
- 4: Absolute Stability (Section 3.3), Stiffness (Section 3.4)
- 5: Backward Euler, Functional Iteration, Newton's method (Section 3.4)
- 6: A-stability, stiff decay (Section 3.5), trapezoidal method (Section 3.6), midpoint method, discontinuities (Section 3.7)
- 7: Review, Intro to Higher-Order Methods, Newton form of the Interpolating Polynomial (p. 125)
- 8: Exam 1
- 9: Exam 1 solutions, Multi-step methods: derivation of variable-stepsize Adams (Section 5.1.1, plus extra material)
- 10: Derivation of variable-stepsize BDF methods (Section 5.1.2, plus extra material), Error of variable-stepsize methods
- 11: Implementation of Linear Multistep Methods (Sections 5.4 and 5.5)
- 12: Order, 0-stability and absolute stability for Linear Multistep Methods (Sections 5.2 and 5.3)
- 13: Intro to Runge-Kutta methods (Sections 4.0-4.2)
- 14: Convergence, 0-stability, order (4.3), and absolute stability for Runge-Kutta methods (Section 4.3)
- 15: Error Estimation and stepsize control for Runge-Kutta methods (Section 4.5, omitting the Step Doubling subsection)
- 16: Implicit Runge-Kutta methods (Section 4.7)
- 17: Review
- 18: Exam 2
On-line
documents
Linda R. Petzold