One step of the method can be written like this:
If we set , we get:
Therefore we obtain:
The region of absolute stability is defined as
Inserting our function, we obtain the following condition:
As we know that , we have:
In order for the proposed method to be stable, m must fulfill the above criteria
Wavenumber Analysis wurde in 2007 nicht behandelt.
The trapezoidal rule is
The rectangle rule is
Simpson's rule is
We can combine the trapezoidal rule and the rectangle rule thus:
This is Simpon's rule.
1 - Diagram
destination | vector of boat
/ <- vector of current
2 - Motion of the boat
We want to find a set of ODEs for
with the speed
We know that there is a constant component due to the current, , and that the rowing of the professors is a vector directed towards the central point with an amplitude of .
This means that at the point
the rowing should be normalised (as it points to the origin) :
and have a length of 10:
This is the speed vector produced by the rowing of the professors.
If we add the constant vector due to the current, we obtain the following coupled system:
3 - Two Euler steps
The explicit Euler method is:
Here the initial position is and we use the time step
First Euler step
The position after ten seconds is .
Second Euler step
Bem.: Entweder keine Taschenrechner waren erlaubt oder ich vergiss meine mitzunehmen :)
The boat is predicted to be at the point after 20s, and it's distance from the parking spot will be
Based on the three data points (t, d) , we could use a 2d. order interpolation polynomial (such as a Lagrange polynomial) passed through all 3 points to estimate when the boat will reach the parking point.
First a little remark:
- means at time j and position i.
So first we do some taylor expansions:
Next we replace the taylor expansions in :
If and :
- Our result for is an exact solution for the heat equation.