# Germund Dahlquist's classical papers on Stability Theory

Electronics - Devices

Oct 8, 2013 (4 years and 9 months ago)

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GermundDahlquist's
classicalpapers
onStabilityTheory
p.1/34
GermundDahlquist's
classicalpapers
onStabilityTheory
GerhardWanner
p.1/34
GermundDahlquist's
classicalpapers
onStabilityTheory
GerhardWanner
Youknow,Iamamultistepman...anddon'ttellanybody,but
therstprogramIwrotefortherstSwedishcomputerwasa
Runge-Kuttacode...
(
G.Dahlquist1982,after10glassesofwine
)
p.1/34
GermundDahlquist's
classicalpapers
onStabilityTheory
GerhardWanner
Youknow,Iamamultistepman...anddon'ttellanybody,but
therstprogramIwrotefortherstSwedishcomputerwasa
Runge-Kuttacode...
(
G.Dahlquist1982,after10glassesofwine
)
Mr.Dahlquist,whenisthespringcoming?
Tomorrow,attwoo'clock.
(
Weatherforecast,Stockholm1955
)
p.1/34
1.FirstDahlquistBarrier(1956,1959).
Thisworkmustcertainlybeconsideredasoneofthegreat
classicsinnumericalanalysis
(
Å.Björk,C.-E.Fröberg1985
).
p.2/34
p.3/34
Themainresultisrathernegative(Thm.4),buttherearenew
formulasofthisgeneralclasswhichareatleastcomparable,.
(
G.Dahlquist1956.
)
p.4/34
Proof.
p.5/34
p.6/34
Thirtyyears
later...
p.7/34
p.8/34
...andwhatcanthismodernketchupbookdobetter..?
p.8/34
p.9/34
ithas
y
n+2
+4y
n+1
−5y
n
=h(4f
n+1
+2f
n
).
.0.51.0
1
2
3
h = 0.1
h = 0.05
h = 0.05
h = 0.025
h = 0.025
h = 0.1
h = 0.05
h = 0.05
h = 0.025
h = 0.025
h = 0.1
h = 0.05
h = 0.05
h = 0.025
h = 0.025
h = 0.1
h = 0.05
h = 0.05
h = 0.025
h = 0.025
h = 0.1
h = 0.05
h = 0.05
h = 0.025
h = 0.025
h = 0.1
h = 0.05
h = 0.05
h = 0.025
h = 0.025
p.9/34
p.10/34
ithas
ρ(ζ)=α
k
ζ
k

k−1
ζ
k−1
+...+α
0
σ(ζ)=β
k
ζ
k

k−1
ζ
k−1
+...+β
0
.
p.10/34
p.11/34
ithas
ζ=
z+1
z−1
orz=
ζ+1
ζ−1
R(z)=(
z−1
2
)
k
ρ(ζ)=
k
￿
j=0
a
j
z
j
,
S(z)=(
z−1
2
)
k
σ(ζ)=
k
￿
j=0
b
j
z
j
p.11/34
p.12/34
ithas
R(z)(log
z+1
z−1
)
−1
−S(z)=C
p+1
(
2
z
)
p−k
+O((
2
z
)
p−k+1
)forz→
p.12/34
p.13/34
ithas
(log
z+1
z−1
)
−1
=
z
2
−µ
1
z
−1
−µ
3
z
−3
−µ
5
z
−5
−...
p.13/34
p.14/34
µ
2j+1
=−
1
2πi
￿
1
−1
x
2j
[(log
1+x
1−x
+iπ)
−1
−(log
1+x
1−x
−iπ)
−1
]dx
=
￿
1
−1
x
2j
[(log
1+x
1−x
)
2

2
]
−1
dx>0.
...andonecandonothingbetter...
p.15/34
µ
2j+1
=−
1
2πi
￿
1
−1
x
2j
[(log
1+x
1−x
+iπ)
−1
−(log
1+x
1−x
−iπ)
−1
]dx
=
￿
1
−1
x
2j
[(log
1+x
1−x
)
2

2
]
−1
dx>0.
0
11

p.15/34
µ
2j+1
=−
1
2πi
￿
1
−1
x
2j
[(log
1+x
1−x
+iπ)
−1
−(log
1+x
1−x
−iπ)
−1
]dx
=
￿
1
−1
x
2j
[(log
1+x
1−x
)
2

2
]
−1
dx>0.
0
11

Althoughthereexistmanydifferentproofsforthetheoremthe
originalpublishedproofstillappearsveryelegant,...
(
R.Jeltsch,O.Nevanlinna1985
)
p.15/34
2.TheSecondDahlquistBarrier(1963).
p.16/34
Ididn'tlikeallthesestrong,perfect,absolute,
generalized,super,hyper,completeandsoonin
mathematicaldenitions,Iwantedsomethingneutral;and
havingbeenimpressedbyDavidYoung'spropertyA,Ichose
thetermA-stable.
(
G.Dahlquist,in1979
).
thefamousdenition...
p.17/34
...andthefamoustheorem
p.18/34
...andthefamoustheorem
...andsomeyearslater...
TalkingonstiffdifferentialequationsinSweden,islike
carryingcoalstoNewcastle...
(
W.L.Miranker,Göteborg1975
).
certainlyoneofthemostinuentialpaperseverpublishedin
BIT
(
Å.Björk,C.-E.Fröberg1985
).
p.18/34
Thesecond
ketchup
p.19/34
Thethird
ketchup
p.20/34
ProofsofDahlquist'sTheorem.
Isearchedforalongtime,nallyProfessorLaxshowedmethe
(
G.Dahlquist,Stockholm1979
,privatecomm.)
p.21/34
ProofsofDahlquist'sTheorem.
Isearchedforalongtime,nallyProfessorLaxshowedmethe
(
G.Dahlquist,Stockholm1979
,privatecomm.)
Thestarswere,however,notreacheduntil1978.
(
anothercitationfromanotherprefaceofanotherspecial
issueofBIT(vol.41,No.5,2001)
)
p.21/34
Originalmotiv.:Ehle'sConj.
(withE.HairerandS.Nørsett)
63036
3
6
6
3
Ehle'sConjecture;OrderStars
63036
3
6
6
3
Ehle'sConjecture;OrderStars
63036
3
6
6
3
Example:BDF2.
3
2
y
n+1
−2y
n
+
1
2
y
n−1
=hf
n+1
y

=λy,µ=hλ⇒(
3
2
−µ)ζ
2
−2ζ+
1
2
=0.
Algebraicequationforζ⇒ζ
1,2
(µ)=

1+2µ
3−2µ
Example:BDF2.

Implicitstage⇒Poleofζ
3
2
y
n+1
−2y
n
+
1
2
y
n−1
=
hf
n+1
y

=λy,µ=hλ⇒
(
3
2
−µ)
ζ
2
−2ζ+
1
2
=0.
Algebraicequationforζ⇒ζ
1,2
(µ)=

1+2µ
3−2µ
Example:BDF2.

Implicitstage⇒Poleofζ

Order⇒e
µ
−ζ
1
(µ)=Cµ
3
+...
Algebraicequationforζ⇒ζ
1,2
(µ)=

1+2µ
3−2µ
Example:BDF2.

Implicitstage⇒Poleofζ

Order⇒e
µ
−ζ
1
(µ)=Cµ
3
+...

A-stable⇒orderstarawayfromimag.axis.
Example:BDF3.
11
6
y
n+1
−3y
n
+
3
2
y
n−1

1
3
y
n−2
=
hf
n+1
(
11
6
−µ)
ζ
3
−3ζ
2
+
3
2
ζ−
1
3
=0.
p.29/34
TheDaniel-MooreConjecture.
Errorconstant.
Jeltsch-NevanlinnaTheorem.
1
1
1
RK2
B
S
scal
1
6⊃S
scal
2
andS
scal
1
6⊂S
scal
2
p.32/34
p.33/34
p.34/34