Part B – Main Part

Fakult¨at f¨ur Mathematik und Informatik 12. Juni 2012 TU Bergakademie Freiberg

W. Queck/M. Helm

Numerical Analysis of Differential Equations Initial Value Problems (V)

============

Part B – Main Part

Exercise 1

The theta method for the differential equationy⁰ =f(t, y) is defined by the equation yn+1=yn+h (1−ϑ)f(tn, yn) +ϑf(tn+1, yn+1)

,

whereϑis a parameter.

a) Show that this method isA-stable forϑ≥ ¹₂.

b) Plot the region of absolute stability for the valuesϑ= 0,¹₄,¹₂,³₄ und 1.

Exercise 2

Determine the region of absolute stability for the midpoint rule y_n+1=y_n−1+ 2hf(t_n, y_n).

Hint: The midpoint rule is a two step method. The region of stability for a multistep method is the set of complex numbersz, for which the zerostof the polynomialπ(t, z) :=ρ(t)−zσ(t) fulfill the root condition.

Exercise 3

Are the one step methods defined by the following Butcher tableausL-stable?

(Additional question:Under which names this methods are also known?)

a) 0 0

1

b) 0 0 0

1 1 0

1/2 1/2

c) 0

1/2 1/2

1/2 0 1/2

1 0 0 1

1/6 1/3 1/3 1/6

d) 1 1

1

e) 0 0 0

1 1/2 1/2 1/2 1/2

f) 1/3 5/12 -1/12

1 3/4 1/4

3/4 1/4

(Hint:It can be shown that the method in (f) is A-stable. This result has not to be proven here.)

Initial Value Problems 2 Exercise 4

Consider the implicit Runge-Kutta method

y^∗ =yn+^h₂f(tn+^h₂, y^∗), yn+1=yn+hf(tn+^h₂, y^∗).

The first step corresponds to the implicit Euler method for the approximation of the value in the midpointtn+^h₂ of the interval, the second step is the midpoint rule which makes use of this value.

a) Determine the region of stability for this method.

b) Is this methodA-stable/L-stable?