Report

Part 1: Implicit Euler scheme and Newton-Raphson method

1.1 To solve for \(c_{i+1}\) using the implicit Euler method, an iterative scheme (in this case: Newton-Raphson) was needed. Briefly explain why this is necessary for the ODE in the assignment.

1.2 The expression for the concentration in the Newton-Raphson scheme is given by: $\(c_{i+1}^{j+1} = c_{i+1}^j - \frac{q(c_{i+1}^j)}{q^\prime(c_{i+1}^j)}\)\( What do the two indices \)i\( and \)j$ refer to?

Part 2: Adams-Bashforth method

2.1 Is the Adams-Bashforth method an implicit or an explicit method? Briefly motivate your answer.

2.2 When you used the Adams-Bashforth method for solving the ODE, you needed to use a different numerical scheme (in your case: explicit Euler) for the first time step. Explain why this is necessary.

Part 3: Numerical stability and accuracy

3.1 In the notebook, you varied the time step for the implicit and explicit Euler scheme and the Adams-Bashforth method. Interpret your results in terms of the stability of the schemes.

3.2 Which numerical scheme would you choose for the ODE in the assignment? Motivate your answer with concrete examples of advantages and disadvantages of the different methods that you experienced during the assignment.

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