Fakultät für Mathematik und Informatik 22. Mai 2013 TU Bergakademie Freiberg
Prof. Dr. O. Rheinbach/Dr. M. Helm
Numerical Analysis of Differential Equations Initial Value Problems (IV)
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 und1.
Exercise 2
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.
Determine the region of stability for this method. Is this methodA-stable?
Exercise 3
Consider once more the initial value problem from session 2 y0=−(y+ 1)(y+ 3), 0≤x≤2, y(0) =−2
with exact solutiony(t) =−3 + 2(1 +e−2t)−1.
a) Determine an approximate solution of the IVP by application of the Adams-Bashforth me- thod (r = 4) with several step sizes and compare with the exact solution. For the startup calculation take the classical Runge-Kutta method (consistency order 4). Compare also with the solutions with Euler modified and the classical Runge-Kutta method from session 2.
b) Observe what happens if the startup calculation is done with Heun’s method (order 3) or the improved Euler method (order 2).
c) Draw a loglog plot of the global error maxi|yi−yexact(ti)| over the stepsize h. Take step sizes from0.25down to2.5·10−5.
Hint: During the next problem session we practise numerical ODE solving with MATLAB. So – if possible of course – please install MATLAB or OCTAVE (http://www.gnu.org/software/octave/, for free) on your laptop and bring it with you for this session.
