**467/667 NUMERICAL METHODS II (3+0) 3 credits**

Numerical differentiation and integration; numerical solution of ordinary differential equations, two point boundary value problems; difference methods for partial differential equations. CS 467 and MATH 467 are cross-listed; credit may be earned in one of the two. Prerequisite(s): MATH 285.

Instructor Course Time Room ---------------------------------------------------------------------- Eric Olson Math 467/667 Numerical Methods II TR 9-10:15am DSMC106

- Instructor:
- Eric Olson
- email:
- Please contact me through WebCampus
- Office:
- DMS 238 and through Zoom by appointment
- Homepage:
- http://fractal.math.unr.edu/~ejolson/467/
- Live Stream:
- If you can't come to class due to sickness, quarantine or other reasons, please join via the Zoom link in WebCampus.

- Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Second Edition, 2008. Note that an online copy of this book is available from the UNR library.

*The Julia 1.6 Language*, official documentation and software download.- Thomas Breloff,
*Plots--Powerful Convenience for Visualization in Julia*. - Jupyter Lab,
*A Comprehensive List of Links for the Juptyer Project*.

- Use Gaussian quadrature to approximate integrals.
- Use Runge-Kutta and Multistep methods to solve IVP's for ODE's.
- Use finite-difference techniques to solve elliptic PDE's.

- Computer Lab 1 Euler's Method (Due Feb 2)
- Computer Lab 2 Multistep Methods (Due Feb 16)
- Computer Lab 3 Interpolating Polynomials (Due Mar 2)
- Computer Lab 4 Explicit Runge-Kutta Methods (Due Apr 6)
- Computer Lab 5 Finite Differences (Due Apr 27)

- HW 1 Exercises 1.1, 1.3, 1.4, 1.5 and optionally 1.7 for extra
credit (Due Feb 21) (solutions).
Hint: Given the way the author has defined the order of a method, checking whether the methods studied in problems 1.3 and 1.5 converge is not necessary. All that's needed is to find the order of the truncation error as O(h

^{p+1}) and then infer that the order of the method is O(h^{p}). Of course no method is useful unless it's convergent. That's just not the focus of these two problems. - HW 2 Exercises 2.1, 2.4, 2.7, 2.9 (Due Mar 7) (solutions).
- HW 3 Exercises 3.1acd, 3.4, 3.7, 3.8 (Due Apr 6) (solutions).
- HW 4 Exercises 4.4, 4.5, 4.6, 4.7 and optionally 4.8 for extra credit (Due Apr 20) (solutions).
- HW 5 Exercises 8.1, 8.3, 8.4, 8.6 (Due May 16) (solutions).

- Lecture 1: Euler's Method
- Lecture 2: Convergence and Order
- Lecture 3: Trapezoid Method
- Lecture 4: Adams-Bashforth Methods
- Lecture 5: General Multistep Methods
- Lecture 6: Dahlquist Root Condition
- Lecture 7: Nystrom Methods
- Lecture 8: Backwards Differentiation
- Lecture 9: Peano Kernel Theorem
- Lecture 10: Orthogonal Polynomials
- Lecture 11: Gaussian Quadrature
- Lecture 12: Explicit RK Methods
- Lecture 13: Implicit RK Methods
- Lecture 14: Stiffness
- Lecture 15: Linear Stability
- Lecture 16: A Stability
- Lecture 17: RK4 and Homework
- Lecture 18: Some Complex Analysis
- Lecture 19: Implicit Method Example (trap-01.jl, trap-02.jl, trap-03.jl)
- Lecture 20: Finite Differences
- Lecture 21: More Finite Differences
- Lecture 22: The Poisson Equation
- Lecture 23: Example Computation (fivepoint.jl)
- Lecture 24: Gershgorin Criterion

- Explain the proofs of the following results:
- Convergence of Euler's method.
- Polynomial Interpolation Theorem.
- Gaussian quadrature has order 2ν.

- Perform the following calculations:
- Compute the truncation error.
- Check the root condition for ρ(w).
- Given ρ(w) find σ(w) to maximize the order.
- Check whether an RK method is A-stable.
- Verify identities for finite difference operators.

- Exactly state the Dahlquist Equivalence Theorem.
- Given ρ(w) and σ(w) write down the multistep method.
- Given a multistep method write the down ρ(w) and σ(w).
- Translate an RK tableau to algebraic notation.
- Explain stiffness.
- Define the linear stability domain.
- The pros and cons of implicit versus explicit methods.
- Definitions of E, Δ
_{+}, Δ_{-}, Δ_{0}and Υ_{0}. - Translate a stencil to algebraic notation.

- Plot the graph (t,y(t)) of an approximation of an ODE.
- Given ρ(w) find σ(w) to maximize the order
possibly using the
`TaylorSeries`library. - Find the interpolating polynomial through points
(x
_{i},y_{i}) and evaluate it for a specific value of x. - Verify the order of an ODE solver by comparing the approximation with a known exact solution.
- Convert the tableau of an explicit Runge-Kutta method into code and approximate an ODE.
- Plot the trajectory (x(t),y(t)) or (x(t),y(t),z(t)) of a vector valued ODE in phase space.

- Explain the proofs of the following results:
- Convergence of Euler's method.
- Polynomial Interpolation Theorem.
- Gaussian quadrature has order 2ν.

- Perform the following calculations:
- Compute the truncation error.
- Check the root condition for ρ(w).
- Given ρ(w) find σ(w) to maximize the order.

- Exactly state the Dahlquist Equivalence Theorem.
- Given ρ(w) and σ(w) write down the multistep method.
- Given a multistep method write the down ρ(w) and σ(w).

Do not come to class if you are sick--even if it's something other than COVID-19. If you are subject to quarantine because of exposure to a person who is sick, please stay home.

This section of Math 467/667 is in person. However, I will live-stream our class meetings each day at a link available in WebCampus for those who are sick or unable to attend on a particular day. I will also maintain an online archive of course materials including lecture notes, assignments and other announcements.

Jan 23-Jan 20 Week 1: 1.1-1.2 Euler's Method Jan 30-Feb 03 Week 2: 1.3-1.4 Theta Method (Lab 1) Feb 06-Feb 10 Week 3: 2.1-2.2 Adams Bashforth Feb 13-Feb 17 Week 4: 2.3 Backwards Differences (Lab 2) *** President's Day Monday Feb 20 Feb 21-Feb 24 Week 5: 3.1 Gaussian Quadrature Feb 27-Mar 03 Week 6: 3.2-3.3 Runge Kutta (Lab 3) Mar 06-Mar 10 Week 7: 3.4-4.1 Implicit RK and Stiffness Mar 13-Mar 17 Week 8: Theoretical Exam *** Spring Break Saturday Mar 18 to Sunday March 26 Mar 27-Mar 31 Week 9: 4.2 Linear and A Stability Apr 03-Apr 07 Week 10: 4.3 Stability of RK Methods (Lab 4) Apr 10-Apr 14 Week 11: Computational Exam Apr 17-Apr 21 Week 12: 8.1 Finite Differences Apr 24-Apr 28 Week 13: 8.2 Two-point boundary problems (Lab 5) May 01-May 05 Week 14: 8.3 Higher order methods May 08 Week 15: Review *** Prep Day May 10 *** Final exam Tuesday, May 16 from 7:30-9:30am in DSMC106

Theoretical Exam 30 points Computer Exam 30 points 5 Computer Labs 6 points each 5 Homework Assignments 6 points each Final Exam 80 points ------------------------------------------ 200 points totalExams and quizzes will be interpreted according to the following grading scale:

Grade Minimum Percentage A 90 % B 80 % C 70 % D 60 %The instructor reserves the right to give plus or minus grades and higher grades than shown on the scale if he believes they are warranted.

Last Updated: Sat Jan 21 02:28:21 PM PST 2023