Internship Weekly Reports Draft

Week 1 (Finite Difference Methods and Computational Fluid Dynamics)

June 23, 2022

Assigned tasks:

1. Set up the computer.
2. Complete Chapter 1 of Ref. (1).
3. Get fundamental CFD knowledge.

Highlights of the week:

• A summary of what I have learned so far about FDM from Ch.1.
• My attempts to get more familiar with Python and matplotlib for CFD.
• Random informal notes for CFD.

Progress updates:

1. I have read the first three sections of the first chapter of Ref. (1) thoroughly and skimmed over the remaining two.
2. Solved a couple of examples & exercises.
3. Successfully set up Jupyter Notebook with Python 3 on Ubuntu and created a Python Virtual Environment for Jupyter.
4. Checked out some matplotlib tutorials and did some simple computations and plottings.
5. I have read the first chapter of Ref. (2) and watched some Youtube videos in an effort to gain a basic understanding of CFD.
6. Uploaded all what I have done on GitHub.

In progress:

1. Setting up the computer.
2. Getting fundamental CFD knowledge.

Things I Need Help with

Nothing yet except for simple questions.

The main challenge I faced last week was choosing which reference to look at out of the huge amount of references I have found.

References

1. Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. Leveque.
2. https://github.com/Rasha22j/summer_internship_2022/blob/master/references/3-s2.0-B9780081011270000015-main.pdf
3. Python Programming and Numerical Methods book.
4. https://lorenabarba.com/blog/cfd-python-12-steps-to-navier-stokes/

Feedback

1. You did well by covering the basic knowledge in chapter 1 and doing some exercises and coding. But in your report, you wrote "Higher order derivatives" and "A general approach to deriving the coefficients " two subsections titles without giving any details.

2. It's good what you wrote about CFD lessons, but please try to answer the questions that I raised in the plan for the research and timeline document:

   1. What is Computational Fluid Dynamics?
   2. Why use Computational Fluid Dynamics?
   3. What are the advantages and disadvantages of Computational Fluid Dynamics?
   4. How do I create a CFD model?
   5. What is the difference between compressible and incompressible flow in fluid dynamics?
   6. What are Reynolds and Mach numbers?

By answering these questions, you will get the basic knowledge to help you start running SSDC code.

Week 2 (IVPs for ODEs and Explicit Runge-Kutta)

June 30, 2022

Assigned tasks:

1. Get familiar with coding using python and SSDC solver.
2. Read about the initial value problem for ODEs and the explicit Runge–Kutta method.
3. Do simulations with python to learn to code a Runge-Kutta algorithm to solve ordinary differential equations.

Highlights of the week:

• A summary of what I have learned about the heat equation from Ch.2.
• My attempts to do solve the 1D, 2D, and 3D heat equation with Python.
• Random informal revesion for ODEs & PDEs.

Progress updates:

1. I have read the first three sections of the second chapter of Ref. (1) thoroughly.
2. Solved a couple of both ODEs & PDEs examples & exercises.
3. Learned more about Python and revised some MATLAB basics.
4. Watched and studied 5 lectures from http://faculty.washington.edu/sbrunton/me565/.
5. Read the first chapter of Ref. (2).
6. Uploaded all what I have done on GitHub.

In progress:

1. Read about Runge–Kutta method.
2. Setting up the computer.
3. Simulations with Python.

Things I Need Help with

Setting up the computer as well as starting with SSDC.

References

1. Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. Leveque.
2. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems by Ricchard Haberman.
3. http://faculty.washington.edu/sbrunton/me565/.

Feedback

1. In your second report, you studied the main categories of PDEs and how you can know if the PDEs are hyperbolic, parabolic, or elliptic.

2. You did very well by studying heat equations and applying what you learned in week 1, and you coded it using python for 1D, 2D, and 3D.

3. Small note, you wrote in the report "Note that since the boundary conditions are functions of time, this means they will not be changing with time" is that true?

Week 3 (SSDC Solver, Accuracy and Stability)

July 7, 2022

Assigned tasks:

1. Get familiar with coding using python and SSDC solver.
2. Read about the initial value problem for ODEs and the explicit Runge–Kutta method.
3. Do simulations with python to learn to code a Runge-Kutta algorithm to solve ordinary differential equations.
4. Analyze the explicit Runge-Kutta method.
5. Study the accuracy and stability of the method.
6. Change parameters and see what happens.

Highlights of the week:

• A summary of what I have learned so far about Runge-Kutta method (basically from Ch.5 of Ref. (1)).
• My attempts to get familiar with Python ODE solvers.

Progress updates:

1. Studied numerical differentiation and numerical integration (Forward Euler, Backward Euler, and Runge-Kutta method).
2. Did some Python numerical differentiation and numerical integration exercises.
3. Continued studying ODEs and PDEs.
4. Watched and studied 5 lectures from https://www.youtube.com/watch?v=QM0ATZRlbKQ&list=PLMrJAkhIeNNR2W2sPWsYxfrxcASrUt_9j.
5. Uploaded all what I have done on GitHub.

In progress:

1. Getting familiar with SSDC solver and studying the accuracy and stability of the method.

Things I Need Help with

The SSDC.

References

1. Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. Leveque.
2. https://www.youtube.com/watch?v=QM0ATZRlbKQ&list=PLMrJAkhIeNNR2W2sPWsYxfrxcASrUt_9j.
3. Python Programming and Numerical Methods book.
4. Numerical methods for partial differential equations by Bernard Knaepen & Yelyzaveta Velizhanina.

Feedback

1. In the third report, you studied numerical differentiation and numerical integration.

2. You wrote subsection Runge-Kutta integration of ODEs and the Lorenz equation without giving any details.

3. In addition, you coded the Runge-Kutta Method for second and fourth ODEs. Well done!

Week 4 (Performing a convergence study and analysing ERK method)

July 21, 2022

Assigned tasks:

1. Continue the analysis of the explicit Runge-Kutta method.
2. Verify the efficiency and accuracy of the method by performing a convergence study.

Highlights of the week:

• Further notes on RK.
• RK stability illustration.
• A simple plot for the convergence study.

Progress updates:

1. Continued studying RK.
2. Ran a Python code to obtain error values and thus plot them versus dt to eventually get a convergence study figure of the optimized ERK method.
3. I have read the paper.
4. Started working on the poster.
5. Uploaded all what I have done on GitHub.

In progress:

Working on the ePoster and the simulation.

Things I Need Help with

This week has been full of challenges; the first one is that I got sick, which kept me from getting much work done. Further, I got some issues with both my email and workstation, which is, in turn, another challenge.

References

1. Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. Leveque.
2. Numerical methods for partial differential equations by Bernard Knaepen & Yelyzaveta Velizhanina.

Feedback

Week 5&6

Aug 4, 2022

During these two weeks, I was primarily working on the e-poster for the competition.

Highlights:

• First place winner :).