==== Part A – to be prepared before the date of problem session =====

Fakult¨ at f¨ ur Mathematik und Informatik 14. Mai 2012 TU Bergakademie Freiberg

W. Queck/M. Helm

Numerical Analysis of Differential Equations Initial Value Problems (III)

==== Part A – to be prepared before the date of problem session =====

Exercise 1

Determine the characteristic polynomials ρ(ζ) und σ(ζ) for the following linear multistep methods.

Verify the conditions

r

X

j=0

α

= 0 and

r

X

j=0

jα

=

r

X

j=0

β

for

a) the 3–step Adams–Bashforth method, b) the 3–step Adams–Moulton method, c) 2–step Simpson method.

============ Part B – Main Part ========================

Exercise 2

Determine the order of consistency for

a) the explicit Nystr¨ om method with r = 2 (midpoint rule) y

− y

= 2hf(t

, y

), b) the implicit Nystr¨ om method with r = 2 (Simpson rule)

y

− y

= h 3 n

f (t

, y

) + 4f (t

, y

) + f (t

, y

) o .

Exercise 3

Which of the following linear multistep methods are convergent? If a method is not convergent: is it not consistent, not zero-stable, or neither?

a) y

=

y

+

y

+ 2hf (y

) b) y

= y

c) y

= y

+

h

f (y

) + f (y

) + f (y

) d) y

= −y

+ y

+ y

+ 2h

f (y

) + f (y

) .