Lösungsvorschlag Wissenschaftliches Rechnen Endterm 2006

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Exercise 1

One step of the method can be written like this:

If we set , we get:

Alternative 1

Therefore we obtain:

 

The region of absolute stability is defined as

Inserting our function, we obtain the following condition:

Alternative 2

As we know that , we have:

This implies:





In order for the proposed method to be stable, m must fulfill the above criteria

Exercise 2

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Exercise 3

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.

Exercise 4

1 - Diagram

destination           | vector of boat
     X                v 
               <----____       
                        ----____B   
                               /   
                              /    
                             /   <- vector of current
                            V

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: where


Here the initial position is and we use the time step

First Euler step

  • For x:









  • For y:







The position after ten seconds is .


Second Euler step

  • For x:











Bem.: Entweder keine Taschenrechner waren erlaubt oder ich vergiss meine mitzunehmen :)

  • For y:









The boat is predicted to be at the point after 20s, and it's distance from the parking spot will be


4 - Extrapolation

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.

Exercise 5

First a little remark:

  • means at time j and position i.

Part 1

So first we do some taylor expansions:


Next we replace the taylor expansions in :

Part 2

If and :

  • Our result for is an exact solution for the heat equation.