Beispiel - SAM - ETH Zürich

τ(y 0 ,h) := Φ h h y 0 −Ψ h y 0 = d(y 0 )h p+1 +O(h p+2 ) für h → 0 , ∀y 0 ∈ D . (4.4.52)
Numerische
Mathemtik
Konsistenzfehler→Def. 2.1.11,
(Φ t h ˆ= Evolutionsoperator zuẏ = f h(y))
(4.4.51): Bestimme∆f so, dass Lsg. von ˙ỹ =˜f h (ỹ),ỹ(0) = y 0
˜Φ h hy−Ψ h y = ỹ(h)−y 1 = O(h p+2 ) fürh → 0 . (4.4.53)
Taylorentwicklung umh = 0, vgl. (2.3.25), benutze Dgl. und Kettenregel: Fürh → 0
ỹ(h) = y 0 +
p+1
∑
k=1
h j
j!ỹ(j) (0)+O(h p+2 ) = y 0 +
= y 0 +h˜f h (y 0 )+ 1 2 h2 D˜f h (y 0 )˜f h (y 0 )
p+1
∑
k=1
h j
j!
d j−1
dt j−1˜f h (ỹ(t))
|t=0
+O(h p+2 )
+ 1 6 h3( D 2˜f h (y 0 )(˜f h (y 0 ),˜f h (y 0 ))+D˜f h (y 0 )D˜f h (y 0 )˜f h (y 0 ) ) +···+O(h p+2 )
= y 0 +hf h (y 0 )+h p+1 ∆f(y 0 )+ 1 2 h2 Df h (y 0 )f h (y 0 )
+ 1 6 h3( D 2 f h (y 0 )(f h (y 0 ),f h (y 0 ))+Df h (y 0 )Df h (y 0 )f h (y 0 ) ) +···+O(h p+2 ) ,
R. Hiptmair
rev 35327,
25. April
2011
da “O(h p )-Modifikation” in (4.4.50), z.B.
h 2 D˜f h (y 0 )˜f h (y 0 ) = h 2( Df h (y 0 )+h p D∆f(y 0 ) ) (f h (y 0 )+h p ∆f(y 0 ))
= h 2 Df h (y 0 )f h (y 0 )+O(h p+2 ) .
4.4
p. 513