*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): 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._
See notebook.
**Question 2**
Include a figure of your solution for dt=0.25 s (task 2.3).
_Your figure here._
See notebook.
**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._
The limit was between 0.3 and 0.4.
Note also that the _implicit_ scheme also has issues when the time step becomes too big, however, this is due to the Newton-Raphson scheme not converging; it is not a stability issue. The solution is stable, but it is also _terrible._
## 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._
See notebook.
**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._
See notebook.
**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_
probably between 20 and 200 time steps (with original values of t0 and t_end)
stability threshold for diffusion should be nu*\Delta t/\Delta x^2 = 0.5 (but we forgot the nu in the equation above, so students will report different ratios)
a good answer would have been:
- find a bad dt
- change dx
- find another bad dt
- find that the ratio dt/dx^2 is about the same for both cases
- (we point out in solution that if you *nu it is around 0.5)
Isabel: grade based on the good answer and we will adjust.
**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._
