The first question on this quiz asks some various things
about the local truncation error of the forward Euler method.
I think the easiest way to start is to know what the actual error is,
and then we can base all the other questions off of this.
If we recall, this is actually the error of the forward Euler method,
so this question is true.
Notice that in this answer, we only have a first derivative here,
which is not the case;
we actually use the first derivative's term in the Taylor Expansion,
so this is false.
And notice here, we have the step size, not the step-size squared,
so this is also false.
The local truncation error of the forward Euler method
is dependent on the dynamical landscape.
We can see this by the fact that there's this f prime prime term (f''),
so this question is false.
And the local truncation error of the forward Euler method
is not proportional to the step size:
it's proportional to the step size squared,
as we can see right here, so this question is also false.
Question 2 is whether the finite-precision arithmetic causes truncation error,
and this is false.
Finite precision arithmetic causes roundoff, or cutoff error.
Truncation error comes about by how we approximate the solution
using a Taylor series, and which term of the Taylor series we truncate.
Question 3 asks if finite-precision arithmetic causes roundoff error.
This is true.
Finite-precision arithmetic can either cause roundoff error or cutoff error
depending on the system being used.
Question 4 asks whether observational error can "snowball" over the course of
a numerical solution to an ODE.
This is false. Observational error is something that can be seen as
dirt on your glasses.
It's something that you're getting at the observation level,
but it is not being fed back into the system.
The "snowballing" effect occurs whenever you have some kind of dynamical error,
or numerical error that is fed back into the system at every time step.
Observational errors are something that you're going to see on your glasses,
not something you fed back into the system.
For question 5, the trapezoidal ODE solver has lower error
than both forward and backward Euler.
The trapezoidal method has local truncation error
on the order of h cubed (h^3),
versus forward and backward Euler, which both have error
on the order of h squared (h^2).
Question 6 asks:
why is it a good idea to adapt the time step of an ODE solver on the fly?
And this is to account for the different curvatures that can occur
in different parts of the dynamical landscape in a nonlinear system.