Report for Group Assignment 1.6

CEGM1000 MUDE: Week 1.6, Friday, Oct 11, 2024.

Questions

Part 1: Solving non-linear ODEs

Question 1

How close is your approximation to the exact solution $x=3$ when your initial guess is 0.01? Explain why it takes more iterations to converge when you use this value instead of a value much farther away than the solution.

Write your answer here.

Question 2

Include a figure of your solution for dt=0.25 s (task 2.3).

Your figure here.

Question 3

By trial and error, find the dt limit of stability for the explicit scheme.

Note that an unstable condition is one that increases/decreases unbounded; an inaccurate solution that has not converged close to the "true" value is not necessarily an unstable condition.

Sate the stability limit here.

Part 2: Diffusion equation in 1D

Question 4

Add an image of the stencils and the algebraic expression of the differential equations for both solution methods: central difference in space with forward and backward difference in time.

Insert image here.

Question 5

Add an image (or Latex equation) of your matrices $AT=b$ for both solution methods. Describe the differences in a few short sentences.

Your answer here.

Question 6

Add an image of the results corresponding to Task 3.8 at t=1500 sec and at t=10000 sec.

Insert image here.

Question 7

From your results of task 3.4 you can observe a dependency on the parameter $\nu \Delta t / \Delta x^2$. Vary $\Delta t$ until you find the stability limit of the Explicit scheme (also print the parameter $\Delta t / \Delta x^2$). What is its value? Now, define $\Delta x$ by half (0.01 instead of 0.02) and vary $\Delta t$ until you find its stability limit and print the parameter $\Delta t / \Delta x^2$. Are the values similar? What is the implication for the computational time?

Your answer should include a couple sentences as an explanation, as well as the values of $\Delta t$ at the limit of stability and the computation time for each approach (see last task of WS 1.6 solution for an example of tracking computation time in Python).

Write your answer here

Question 8

For the implicit scheme, try to find a $\Delta t$ value for which the solution is not reasonable. State your result and explain.

Write your answer here

Question 9

Considering the non-linear ODE and the PDE results, would you say that Implicit methods are always better than Explicit methods? State "yes" or "no" and provide a brief explanation (2-3 sentences).

Insert image here

Last Question: How did things go? (Optional)

Use this space to let us know if you ran into any challenges while working on this GA, and if you have any feedback to report.

End of file.