Scientific Computing - Quiz 2 - Question 3-6

Problem:

Consider the differential equation
$ \frac{dy}{dt} = y $

with y(0)=1. The exact solution to this problem is given by $ y(t)=e^t $. Knowing the exact solution and using $\Delta t=0.1$ with the Euler scheme, compute the local error for two iterative steps ((a) $\epsilon_1$, (b) $\epsilon_2$) and compute the global error ((c) $E_1$, (d) $E_2$) at each of the three steps.

Solution:

The local error computation is exactly the same as the last problem.

$ \epsilon_1 = e^{0.1} - (1 + 0.1) \approx  0.0052 $
$ \epsilon_2 = e^{0.2} - (e^{0.1} + 0.1e^{0.1}) \approx 0.0057 $

For the 1st step, the global error is actually the same as the local error.

$ E_1 = \epsilon_1 = 0.0052 $

For the 2nd step, we need to actually iterate twice.

$ y_1 = 1 + 0.1 $
$ y_2 = y_1 + 0.1 y_1 = 1.21 $

Therefore $ E_2 = e^{0.2} - 1.21 \approx 0.011 $

Notice an important observation here - $ \epsilon_2 \neq E_2 - E_1 $. The error introduced in the last step also lead to wrong slope estimation, the local error of the second step assumed the correct slope will be there!