# CHEN 2450 – Numerical Methods – Spring 2020

Phone Number: 801 585 0344
Class:
Tu/Th 2:00 PM – 3:20 PM – WEB 2250
Office Hours: Tuesdays from 3:30 PM to 4:30 PM or by appointment
Office Location: CME 115

## Course Catalog Description

Applications of numerical methods to interpolation, differentiation, integration, and the solution of systems of linear, nonlinear, and differential equations in chemical engineering.

## Learning Outcomes and Topics:

The primary goal of this course is to provide engineering students with a basic working knowledge of numerical methods. In addition, this course aims at developing an appreciation of programming and how computers can be an asset for the modern-day engineer.

My goals in this class are to:

1. Teach you the importance of numerical methods
2. Introduce you to the skills needed by modern engineers
3. Emphasize the importance of knowing a few programming languages
4. Emphasize learning and critical thinking

By the end of this class, students will have a working knowledge of the following:

• Errors
• Absolute and relative errors
• True and approximate error
• Sources of numerical errors: Roundoff, truncation, and iterative errors
• Solution of linear systems of equations
• Direct and iterative solution methods
• Sparse matrices
• Pivoting and diagonal dominance
• Tridiagonal systems (TDMA)
• Jacobi, Gauss-Seidel, and SOR
• Interpolation
• Linear, polynomial, and cubic splines
• 2D interpolation
• Regression
• Linear and nonlinear least-squares regression
• Solution of nonlinear equations
• Open and closed domain methods
• Newton’s method
• Systems of nonlinear equations
• Numerical differentiation
• Numerical integration
• Newton-Cotes integration: midpoint, trapezoid, and Simpson’s rules
• Numerical integration of discrete data
• Solution of ordinary differential equations (ODEs)
• Initial and boundary value problems
• Explicit and implicit methods
• Basic ODE stability
• Systems of ODEs
• Finite difference methods
• Introduction to PDEs

The following are listed as prerequisites and co-requisites for ChEn 2450:

• Prerequisites:  ChEn 1703,  Algebra & calculus
• Corequisite: Math 2250 – Ordinary differential equations & Linear Algebra

There is no textbook for this class. Material presented in this class will be based on the following resources:

• Chapra, S. C., & Canale, R. P. (2010). Numerical methods for engineers , 7th edition. Boston: McGraw-Hill Higher Education

## Lecture Notes

 # Date Lecture Notes (PDF/Web) Goals Screencast Other HW 1 Tuesday, 01/07/2020 1. Introduction.pdf 2. Python Primer Introduction, Jupyter notebook, Python. 1. Introduction 2. Why Python? Jupyter guide 2 Thursday, 01/09/2020 1. EasyPy 2. Errors.pdf Python tutorial. Measuring errors, true errors, absolute and relative errors. 1. EasyPy 2. Errors 1 HW1 Assigned 3 Tuesday, 01/14/2020 Approximate error, Sources of error, Roundoff, Truncation, and Iterative errors. 1. Errors 2 4 Thursday, 01/16/2020 Linear Solvers.pdf 1. Finish off discussion on errors. 2. Introduction to systems of linear equations, sparse matrices. 1. Errors 3 2. Linear Solvers 1 Linear algebra refresher HW2 Assigned 5 Tuesday, 01/21/2020 1. Go over homework 1 2. Continue discussion on the TDMA 3. Heat transfer example for TDMA 1. HW1 Solution 2. Linear Solvers 2 6 Thursday, 01/23/2020 1. Continue with the heat transfer example 2. Start on Iterative Solvers 1. Heat Transfer Example 2. Iterative Solvers 1 HW3 Assigned 7 Tuesday, 01/28/2020 1. Solve HW2 2. Iterative Solvers 2: Error Norms 1. HW2 Sol 2. Iterative Solvers 2 8 Thursday, 01/30/2020 1. Gauss-Seidel, SOR 2. Sparse matrix storage 3. Iterative Solvers in Python HW4 Assigned 9 Tuesday, 02/04/2020 Iterpolation.pdf 1. Interpolation 1 1. Interpolation 1 (video) 10 Thursday, 02/06/2020 1. Interpolation 2 1. Interpolation 2 (video) HW5 Assigned 11 Tuesday, 02/11/2020 Regression.pdf 1. Regression 1 1. Regression 1 (video) Least Squares Example 1.ipynb 12 Thursday, 02/13/2020 1. Go over HW 4 2. Continue Regression (no audio on screencast – sorry) Tuesday, 02/18/2020 1. Solve HW5 2. Exam 1 Review 2. Regression 3 Regression 3 (video) 14 Thursday, 02/20/2020 Exam 1 Study Guide.pdf Exam 1 Review.pdf Practice Exam 1.pdf Exam 1(Errors, Linear Systems, and Interpolation) 15 Tuesday, 02/25/2020 Regression Regression 4 16 Thursday, 02/27/2020 Regression (no audio on screencast – sorry) HW 6 Assigned 17 Tuesday, 03/03/2020 Integration.pdf Numerical Integration 18 Thursday, 03/05/2020 Differentiation.pdf Numerical Differentiation HW 7 Assigned 19 Tuesday, 03/10/2020 NO CLASS – SPRING BREAK NO CLASS – SPRING BREAK SPRING BREAK SPRING BREAK 20 Thursday, 03/12/2020 NO CLASS – SPRING BREAK NO CLASS – SPRING BREAK SPRING BREAK SPRING BREAK 21 Tuesday, 03/17/2020 NO CLASS NO CLASS NO CLASS NO CLASS 22 Thursday, 03/19/2020 Nonlinear Solvers.pdf Intro to nonlinear solvers (Video) The Bisection Method (video) The Method of False Position (video) The Secant Method (video) Newton’s Method (video) Root Finding Methods.ipynb 23 Tuesday, 03/24/2020 Systems of Nonlinear Equations (video) Nonlinear System Demo.ipynb 24 Thursday, 03/26/2020 Exam 2 Review and other items 25 Tuesday, 03/31/2020 Exam 2 study guide.pdf Exam 2 review.pdf Practice exam 2.pdf EXAM 2 (Regression, Integration & Differentiation, Nonlinear Solvers) 26 Thursday, 04/02/2020 Ordinary Differential Equations.pdf ODEs 1 (video) Explicit Time Integration.ipynb Motivation for Implicit Methods.ipynb Cyclical Systems.ipynb System of Equations – Kinetics Example.ipynb 27 Tuesday, 04/07/2020 ODEs 2 (video) 28 Thursday, 04/09/2020 ODEs 3 (video) HW8 Assigned 29 Tuesday, 04/14/2020 ODEs 4 30 Thursday, 04/16/2020 ODEs 5 31 Tuesday, 04/21/2020 Wrap up and Review 32 Friday, 04/24/2020 FINAL EXAM FINAL EXAM FINAL EXAM FINAL EXAM

See here for fall semester schedule: https://registrar.utah.edu/academic-calendars/spring2020.php

Event Date
Semester Length Classes
Classes begin Monday, January 6
Last day to add without a permission code Friday, January 10
Last day to wait list Friday, January 10
Last day to add, drop (delete), elect CR/NC, or audit classes Friday, January 17
Last day to withdraw from classes Friday, March 6
Last day to reverse CR/NC option Friday, April 17
Classes end Tuesday, April 21
Final exam period Thurs-Wed, April 23-April 29
• Getting Help: Meeting times: coming soon
• Discussion sections: coming soon
• Teaching Assistants: coming soon
• College of engineering guidelines discusses withdrawal policies, ADA policies, etc.
• Attend the discussion section you are registered for.  You may attend other ones in addition for extra help.
• Use the course website, lectures, and online resources.
• If all the above fails, then feel free to stop by and we can discuss any gaps in your understanding of the subject matter.

## Python

This class will make exclusive use of Python – a modern programming language that is suitable for scientific computing. Python is easy to use and  – most importantly – free!

Please go here for an easy tutorial on Python by Prof. Saad. In addition, Prof. Saad will hold a few in-class lectures on learning Python.

###### Accessing Matlab

There are several options for accessing MATLAB.

• The most convenient for ChEn students is probably via the ICC (MEB 2285), which is a Chemical Engineering computer lab.  To set up an account, follow this link.  You can also gain remote access to this lab from any computer with a (fast) network connection.
• You can purchase the student version of MATLAB.  There are also several free MATLAB alternatives, including Octave and FreeMAT.  These don’t have all of the features of MATLAB, but are probably sufficient for what you will need in this class.
• The CADE lab in WEB has Windows (WEB 210), Mac (WEB 210) and Linux (WEB 246) computers. Walk in and find one of the system administrators to get set up with an account.  Also, you can access the Linux machines remotely if you have a fast internet connection.  Only try this if you are familiar with X-windows and SSH.
• Library computers running Windows.  (I don’t think that MATLAB is installed on the Mac computers in the library)

## Homework

• Homework is a fundamental piece of the learning process. It will help you strengthen the concepts you learned in class and apply them to new problems.
• The goal of homework is to get you to familiarize yourself with the nomenclature and the types of problems that can be solved with numerical methods.
• Homework assignments will be posted on Canvas. Unless otherwise stated, homework is due by the beginning of class on the date indicated on the schedule.
• Solutions will be discussed in class the day after the homework is due.
• Feel free to “work together” on homework assignments. Discuss the various solutions methods and attempt to learn or fill deficits in your understanding of the subject matter. However, you must submit your own original work. Please do not cross the line of plagiarism and cheating. Such behavior will not be tolerated.
• Homework assignments must be submitted electronically via Canvas. Your report should be submitted as a single Jupyter notebook (.ipynb) that combines both a reasonable discussion of your results as well as your code. If your code does NOT run, it will be assumed that it does not work.

## In-Class comprehension quizzes

Every now and then, I will give you a short “comprehension” quiz. The quiz should take no more than five minutes to complete and will address the comprehension aspect of numerical methods. For example: What do you use interpolation for? Give an example of where linear systems arise?

The quiz will also challenge your critical thinking. For example, if the temperature at 8 AM was 25 degrees and at 9 AM it was 35 degrees. An engineer estimated the temperature at 8:30 AM to be 55 degrees. Does this make sense?

These quizzes are aimed to test your understanding of the subject matter in “words”.

• 20% each midterm exam (two midterms)
• 25% Homework
• 10% In-Class comprehension quizzes
• 25% Final exam

Grades will be assigned on the following scale, normalized to the highest student in the class (who, by definition, is 100%)

• 92< A ≤ 100,   89 < A- ≤ 92
• 86 < B+ ≤ 89,   81 < B ≤ 86,   78 < B- ≤ 81
• 75 < C+ ≤ 78,   70 < C ≤ 75,   67 < C- ≤ 70
• 64 < D+ ≤ 67,   59 < D ≤ 64,   56 < D- ≤ 59
• E ≤ 56

I reserve the right to adjust this scale downward if I deem it necessary.