Scientific Computing - Quiz 1 - Question 5

Problem:

Using Eqs.~(1.1.10)-(1.1.12) found in the lecture notes found under the course resource link, derive the three important schemes: Heun's method (A = 1/2), modified Euler-Cauchy (A = 0) and Euler (A = 1).

Solution:

Here is the link to the full lecture notes:

The derivation based on 1.1.12 is pretty easy, for Heun's method:

$ A + B = 1 \implies B = \frac{1}{2} $.
$ PB = \frac{1}{2} \implies P = 1 $.
$ BQ = \frac{1}{2} \implies Q = 1 $.

For Euler-Cauchy method:

$ A + B = 1 \implies B = 1 $.
$ PB = \frac{1}{2} \implies P = \frac{1}{2} $.
$ BQ = \frac{1}{2} \implies Q = \frac{1}{2} $.

For Euler method:

$ A + B = 1 \implies B = 0 $.
P and Q are meaningless in the context of the iteration, we do not want any correction at all in this case.