Math 467/667

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

Course Information

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.

Required Texts

1. 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.

Information about Software

• 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.

Student Learning Outcomes

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

Computer Labs

• 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)

Homework

• 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 Notes

• 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

Announcements

[20-May-2023] HW5 Solutions

[16-May-2023] Final Exam

• 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_-, Δ₀ and Υ₀.
• Translate a stencil to algebraic notation.

[1-May-2023] HW4 Solutions

[27-Apr-2023] In-class Computer Lab 5

[13-Apr-2023] Computational Exam

• 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.

[8-Apr-2023] HW3 Solutions

[6-Apr-2023] In-class Computer Lab 4

[16-Mar-2023] Theoretical Exam

• 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).

[13-Mar-2023] HW2 Solutions

[2-Mar-2023] In-class Computer Lab 3

[26-Feb-2023] HW1 Solutions

[16-Feb-2023] In-class Computer Lab 2

[2-Feb-2023] In-class Computer Lab 1

[21-Jan-2023] Welcome Spring 2023

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.

Tentative Course Schedule

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

Grading

     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 total

    Grade        Minimum Percentage
      A                 90 %
      B                 80 %
      C                 70 %
      D                 60 %

Quiz and Exam Schedule

Course Policies

Communications Policy

Late Policy

Plagiarism

Diversity

Statement on Academic Success Services

Equal Opportunity Statement

Statement on Audio and Video Recording

Academic Conduct

Final Exam