Report

Part 1

1.1 (0.5 points)

Classify the model equation based on the categories listed under chapter 2.1 of the MUDE textbook (paste your answer from task 1.1).

Part 2

2.1 (0.5 points)

Write down the discretized equation for \(S_{i+1}\) based on the implicit Euler scheme (paste your result from task 2.1). Explain why an iterative scheme is needed to solve for \(S_{i+1}\).

2.2 (1 point)

Write down the function \(g(z)\) whose root is solved with Newton–Raphson, and specify what \(z\) represents in this context (use your solution from task 2.2).

2.3 (0.5 point)

Derive and present the expression for \(g'(z)\) used in the Newton–Raphson update for the ODE of the conceptual hydrological model (use your answer from task 2.3).

2.4 (0.5 point)

Provide the full expression used for iteration in the Newton-Raphson scheme, \(S_{i+1}^{j+1}\) (use your answer from task 2.3).

Part 3

3.1 (1 point)

Use a time step of 0.1 days, a tolerance of \(10^{-4}\) and a maximum number of Newton iteration of 50. What are the first five values of the time \(t\) and the storage \(S\) obtained with your implicit Euler scheme? Include at least four decimal digits.

3.2 (1 point)

Use a time step of 1 day, a tolerance of \(10^{-4}\) and a maximum number of Newton iteration of 50. Paste the plot from task 3.1 that compares your implicit Euler solution to the solution obtained with scipy.

How do the two solutions differ? How can you explain these differences?

3.3 (1 point)

How does the tolerance relate to the number of Newton iterations needed? What can you conclude from this on the efficiency of the Newton scheme? (Use your results from task 3.3.)

3.4 (2 points)

Reflect on the goodness of fit for the simulated discharge time series, including any temporal patterns in the deviations. What reasons could explain the misfit? What changes would you propose to improve the model?

3.6 (1 point)

To get a more realistic representation of hydrologic processes, many models include several states. (For example, these states could represent a fast flow and slow flow component, or snow storage, but this is irrelevant for this question.) If you want to include such additional states in the model, how would this affect the numerical solution scheme and its computational cost (still using an implicit Euler scheme with Newton-Raphson iterations)?

By Anna Störiko, Ronald Brinkgreve, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook.