Report

Report#

Please be concise, but include essential elements such as analyses, figures, and explanations that support your answers to the questions.

Part I#

Question 1 (1 pt)

Which expression did you reach for the truncation error of the Crank-Nicolson scheme?

Copy here your answer of Task 1.1. Including the mathematical formulations is sufficient, but you may want to add comments on what was done for clarity.

Question 2 (0.5 pt)

Given the test equation

\[ \frac{dy}{dt} = \lambda y \]

Why does \(\lambda\) have to be negative? Include in your answer what happens to the solution when \(\lambda > 0\) and \(\lambda < 0\).

Write your answer here.

Question 3 (1 pt)

Provide a brief explanation with two figures from Tasks 1.3 and 1.4 on how you arrived at the final time steps of both Crank-Nicolson and forward Euler schemes in such a way that they are sufficiently accurate. Base your explanation on the concept of order of accuracy. Include any relevant quantitative values and graphs.

Part II#

The following five questions are related to Task 2.1 of the notebook. You may copy your answers from this task directly.

Question 4 (0.5 pt)

Which of the schemes has the most numerical diffusion?

Question 5 (0.5 pt)

For what value of \(\theta\) is the numerical diffusion the largest?

Question 6 (0.5 pt)

For what value of \(\theta\) is there no numerical diffusion?

Question 7 (0.5 pt)

Which of the two contributions to numerical diffusion has the largest impact?

  • the first order upwind scheme or

  • the \(\theta\)-method

Question 8 (0.5 pt)

Which of the schemes gives a non-negative solution?

Part III#

Question 9 (1 pt)

Provide here a figure of the results according to Task 3.1. In your answer, comment on the cases without and with diffusion include the corresponding values of \(K\). What is the mesh Péclet number?

Question 10 (1 pt)

Provide here two figures from Tasks 3.2 and 3.3, respectively.

Derive the stability limit for the FTCS scheme. Show your derivation!

Provide here your explanation and conclusions based on your findings. Include any calculations you made to reach your conclusions, and any recommendations for modifying any parameters, if at all.

Question 11 (1 pt)

Provide here two figures from Tasks 3.4 and 3.5, respectively.

Derive the stability limit for the FTBS scheme. Show your derivation!

Provide here your explanation and conclusions based on your findings. Include any calculations you made to reach your conclusions, and any recommendations for modifying any parameters, if at all.

Question 12 (1 pt)

Provide a figure that shows three stable numerical solutions of the advection-diffusion equation with \(v= 0.35\) m/s and \(K = 55\) m2/s, namely,

  • the one obtained from Crank-Nicolson with central differences (Task 3.1),

  • the one obtained from stable FTCS scheme (Task 3.3), and

  • the one obtained from stable FTBS scheme (Task 3.5).

Answer the following questions.

  1. Which of the schemes is the most accurate one? Why?

  2. Which of the numerical solutions has the largest maximum? Why? You may include numerical values to help illustrate your answer.

End of file.

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