SIAM
J.
NUMER.
ANAL.
Vol.
33,
No.
4,
pp.
1425-1450,
Augus
1996
()
1996
Socie y
o
Indus ial
and
Applied
Ma hema ics
008
AN
EFFICIENT
TWO-DIMENSIONAL
VORTEX
METHOD
WITH
LONG
TIME
ACCURACY*
IBRAHIM
BLESS
RANERO
AND
TOM/S
CHAC(3N
REBOLLO
Abs ac .
This
pape
deals
wi h
e icien
echniques
o
he
nume ical
solu ion
o
wo-dimensional
ee-space
incomp essible
Eule
equa ions.
We
de elop
an
algo i hm
o
as
compu a ion
o
eloci y
in
a
o ex
me hod
based
upon
disc e iza ion
o
o ici y
by
ini e
elemen s.
We
p o e
ha
he
me hod
wi h
as
compu a ion
o
eloci y
is
nume ically
s able
and
con e gen
wi h
second-o de
accu acy.
Some
s anda d
nume ical
es s
show
ha
he
algo i hm
wi h
Delaunay
eg idding
bea s
good
s abili y
and
accu acy
p ope ies
o
long
in eg a ion
imes,
wi h
a
ela i ely
low
compu a ional
cos .
Mo eo e ,
he
algo i hm
is
ound
o
be
mo e
accu a e
han
high-o de
o ex-blob
me hods
wi h
eg idding
o
long
enough
in eg a ion
imes.
Key
wo ds,
o ex
me hods,
ini e
elemen s,
Delaunay
eg idding,
long
ime
accu acy
AMS
subjec
classi ica ion.
65N30
1.
In oduc ion.
This
pape
deals
wi h
he
nume ical
solu ion
o
wo-dimensional
ee-
space
incomp essible
Eule
equa ions
by
means
o
o ex
me hods
wi h
ini e
elemen s.
Lag angian
me hods,
and
in
pa icula
o ex
me hods,
ha e
many
a o able
p ac ical
p ope ies
o
he
nume ical
simula ion
o
incomp essible
lows
a
a
high
Reynolds
numbe
(c .
Leona d
18],
Majda
19]
o
a
de ailed
bibliog aphy
and
Ande son
1
o
lows
in
bounded
domains).
Vo ex
me hods
essen ially
in oduce
no
nume ical
iscosi y
and
a e
qui e
accu a e
and
s able,
a
leas
o
sho
imes
(c .
Beale
and
Majda
[5],
Pe lman
[20]).
A
comple e
heo y
o
s abili y
and
con e gence
o
o ex
me hods
o
ee-space
wo-
and
h ee-dimensional
Eule
equa ions
has
been
de eloped
since
he
la e
se en ies.
In
Ande son
and
G eenga d
[2]
and
Beale
and
Majda
[3],
[4]
an
analysis
o
con e gence
in/P-no ms,
wi h
ini e
p,
may
be
ound.
La ely,
a
heo y
o
con e gence
in/-no ms
was
de eloped,
mos ly
by
Hou
and
his
collabo a o s
(c .
Hou
and
Loweng ub
[16],
Hou
[14],
Hou
[15]).
Those
wo ks
p o e
ha
o ex
me hods
a e
essen ially
s able
in
p-
and/-no ms,
bu
s abili y
is
condi ioned
o
a
ela i ely
high
o de
o
consis ence.
Such
"condi ioned
s abili y"
appea s
in
all
con e gence
p oo s
o
o ex
me hods,
as
essen ially
ela ed
o
he
singula i y
o
he
B
io -Sa a
ke nel.
In
p ac ice,
o ex
me hods
compu e
qui e
accu a e
solu ions
o
Eule
equa ions
o
ela-
i ely
sho
in eg a ion
imes.
Beale
and
Majda
[5]
and
Pe lman
[20]
ha e
es ed
o ex-blob
algo i hms
wi h
high-o de
ke nels.
Those
expe imen s
con i m
he
heo e ical
p edic ions
o
o de
o
con e gence
o
mode a e
imes;
howe e ,
a
la e
imes
he
high-o de
accu acy
p og essi ely
de e io a es.
This
seems
o
be
due
o
he
p og essi e
inc ease
o
local
g id
size
ha ,
in
gene al,
akes
place
as
he
ini ial
g id
is
de o med
by
he
low.
This
leads
o
an
inc eas-
ing
loss
o
accu acy
in
he
compu a ion
o
eloci ies.
As
s abili y
is
condi ioned
o
enough
accu acy,
a
p og essi e
loss
o
s abili y
also
occu s.
Beale
and
Majda
conclude
ha
o
ob ain
accu a e
solu ions
o
long
in eg a ion
imes,
eg idding
echniques
a e
usually
needed.
In oducing
eg idding
echniques
dec eases
he
local
g id
size
and
allows
s abili y
and
accu acy
o
longe
imes.
Un o una ely,
his
in oduces
inc easing
le els
o
nume ical
di usion,
coun e balancing
he
main
ea u e
o
o ex
me hods.
Ou
pu pose
he e
is
essen ially
o
de elop
a
o ex
me hod
in
which
i
is
possible
o
educe
he
local
g id
size
wi hou
in oducing
nume ical
di usion
a
he
g id
nodes.
Ou
*Recei ed
by
he
edi o s
May
19,
1993;
accep ed
o
publica ion
(in
e ised
o m)
Sep embe
28,
1994.
This
esea ch
was
pa ially
suppo ed
by
Spanish
M.
E.
C.
P ojec
DGICYT
PB91-0619.
Depa amen o
de
Ma emi ica
Aplicada,
Uni e sidad
Complu ense
de
Mad id,
A da.
de
la
Complu ense,
s/n.
20840
Mad id,
Spain
([email p o ec ed]).
Depa amen o
de
Ecuaciones
Di e enciales
y
Amilisis
Num6 ico,
Uni e sidad
de
Se illa,
C/Ta ia,
s/n.
41080
Se illa,
Spain
([email p o ec ed]).
1425
1426
IBRAHIM
BLESS
RANERO
AND
TOM/S
CHACON
REBOLLO
wo k
is
based
upon
a
o ex
me hod
in oduced
in
Chacon
and
Hou
10].
In
his
me hod
he
o ici y
is
disc e ized
by
means
o
piecewise
linea
ini e
elemen s.
Also,
he
mesh
poin s
o
he
ini ial
iangula ion
a e
anspo ed
along
he
s eamlines
o
he
disc e e
low.
Thus,
he
o ici y
is
accu a ely
compu ed
a
he
mesh
poin s,
since
he
o ici y
is
conse ed
along
s eamlines.
This
me hod
sha es
many
nice
p ope ies
o
o ex
me hods,
being
in
pa icula
nondissipa i e.
The
me hod
is
p o ed
o
be
s able
and
con e gen
wi h
second-o de
accu acy
in
he
uni o m
no m.
Howe e ,
he
s abili y
is
condi ioned
o
an
o de
o
accu acy
bigge
han
one,
much
as
in
classical
o ex
me hods.
Nume ical
expe imen s
show
ha
o
sho
in eg a ion
imes
his
me hod
is
qui e
accu a e,
bu
ha
as
ime
inc eases,
he
iangula ions
gene ally
become
degene a ed.
This
p oduces
a
as
loss
o
accu acy
in
a
sho
ime
a e
degene a ion.
On
he
o he
hand,
he
me hod
uses
a
echnique
o
compu a ion
o
disc e e
eloci ies
ha
equi es
an
amoun
o
ope a ions
o
o de
O
(N2),
N
being
he
numbe
o
g id
nodes
in
he
suppo
o
he
o ici y.
These
wo
d awbacks
make
he
me hod
un easible
in
p ac ical
cases.
This
pape
epo s
some
modi ica ions
o
his
me hod
ha
ende
i
accu a e
o
e y
long
imes,
wi h
a
ela i ely
low
compu a ional
cos .
A
i s ,
we
de elop
a
as
echnique
o
compu e
he
disc e e
eloci y
in
he
Chacon-Hou
algo i hm.
This
is
an
adap a ion
o
he
echnique
in oduced
in
G eenga d
and
Rokhlin
12]
o
ou
con ex
o
disc e iza ion
o
o ici y
by
ini e
elemen s.
This
educes
he
amoun
o
compu a ional
wo k
equi ed
by
he
me hod
o
O(N
log
2
N).
We
p o e
an
es ima e
o
e o ,
in
e ms
o
he
uni o m
no m
o
he
o ici y,
o
he
compu a ion
o
disc e e
eloci ies
by
his
echnique.
Fu he mo e,
we
p o e
ha
i
in
he
Chacon-Hou
algo i hm
he
eloci y
is
compu ed
by
his
echnique
wi h
enough
accu acy,
he
nume ical
solu ion
is
s ill
con e gen
in
he
uni o m
no m
wi h
second-o de
accu acy.
The
p oo
deals
essen ially
wi h
he
ac
ha
ou
algo i hm
keeps
cons an
he
uni o m
no m
o
he
disc e e
o ici y.
This
allows
us
o
ob ain
uni o m-
in- ime
es ima es
o
he
e o
in
he
compu a ion
o
he
eloci ies
wi h
he
as
algo i hm
and
ensu e
he
s abili y
o
he
algo i hm.
We
also
p o e
ha
i
Delaunay
eg idding
is
in oduced,
he
algo i hm
wi h
as
compu a-
ion
o
eloci y
is
s ill
con e gen
wi h
second-o de
accu acy.
Delaunay
eg idding
cons uc s
he
iangula ion
ha
maximizes
he
smalles
angle
o
all
possible
iangula ions
suppo ed
by
a
gi en
cloud
o
poin s.
In oducing
his
eg idding
echnique
in
ou
me hod
p oduces
he
only
e ec
o
ede ining
he
connec ions
be ween
g id
poin s.
The
alues
o
he
disc e e
o ici y
a
he
g id
poin s
emain
unchanged.
Thus,
he
local
g id
size
is
educed,
wi hou
in oducing
nume ical
di usion.
The
algo i hm
is
con e gen
independen
o
he
ac ual
ime-s epping
s a egy
used
o
apply
Delaunay
eg idding.
We
inally
epo
some
nume ical
expe imen s
dealing
wi h
es
cases
conside ed
by
Beale
and
Majda.
We
con i m
ou
heo e ical
expec a ions
on
he
numbe
o
ope a ions
in
he
compu a ion
o
disc e e
eloci ies.
We
also
pe o m
some
es s
o
he
modi ied
Chacon-
Hou
algo i hm,
in oducing
Delaunay
eg idding.
These
es s
show
excellen
p ope ies
o
s abili y
and
accu acy,
o
e y
long
ime
in e als,
in
he
es
cases
conside ed.
The
e o s
a e
conse ed
close
o
he
ini ial
alues,
and
he
heo e ical
con e gence
o de s
a e
con i med
nume ically,
e en
o
long
in eg a ion
imes.
We
also
obse e
ha
his
algo i hm
compa es
ad an ageously
o
a
desingula ized
o ex
me hod
o
he
same
o de ,
and
also
o
high-o de
o ex-blob
me hods
wi h
eg idding
o
long
enough
imes
o
in eg a ion.
Ou
pape
is
o ganized
as
ollows.
In
2,
we
desc ibe
he
algo i hm
epo ed
in
Chacon
and
Hou
10].
In
3
we
de elop
he
echnique
o
as
compu a ion
o
eloci ies.
Sec ion
4
is
de o ed
o
he
analysis
o
he
con e gence
o
he
modi ied
Chacon-Hou
algo i hm.
Finally,
in
5
we
epo
ou
nume ical
es s.
LONG
TIME
ACCURATE
VORTEX
METHOD
1427
2.
Desc ip ion
o
he
base
algo i hm.
In
his
sec ion
we
shall
desc ibe
he
o ex
me hod
wi h
ini e
elemen s
in oduced
in
Chacon
and
Hou
10],
as
some
ele an
p ope ies
o
his
me hod
mo i a e
ou
wo k.
We
a e
in e es ed
in
he
nume ical
solu ion
o
ee-space
Eule
equa ions
in
wo
space
dimensions,
wi h
a
homogeneous
condi ion
a
in ini y.
In
o ici y
(co)-s eam unc ion
(q)
o mula ion,
hese
equa ions
a e
CO
-[-
(U
V)co
0
in
Rex]0,
T[,
co(x,
0)
co0(x)
in
R
e,
(1)
-Aq
co
inR
2,
lim
q(x)
0.
He e,
u
(u
l,
u2)
is
he
eloci y
ield,
de ined
by
he
wo-dimensional
Bio -Sa a
law,
(2)
u(x,
)
(K
co)(x)
[
K(x
y)
co(y,
)
dy,
whe e
K
is
he
B
io -Sa a
ke nel,
(3)
g(x)
2z lxl
2
(-x2,
Xl).
Equa ions
(1)
a e
equi alen
o
he
usual
o mula ion
o
Eule
equa ions
(c .
Ande son
and
G eenga d
[2],
Ka o
[1
]).
The
equa ion
o
he
o ici y
in
(1)
may
be
in eg a ed
exac ly
on
he
s eamlines
X
( ;
s,
associa ed
o
he
eloci y
ield
u.
The
cu e
6
[0,
T]
--+
X
( ;
s,
o )
6
R
2
desc ibes
he
ajec o y
o
a
luid
pa icle
whose
posi ion
a
ime
s
is
he
poin
o
6
R
2.
I
sa is ies
he
ollowing
o dina y
di e en ial
equa ion:
-d--( ;
s,
o )
u(X( ;
s,
),
)
in
[0,
T],
(4)
X(s;
,)
=.
Then,
(5)
co(X
( ;
s,
o ),
)
co(o ,
s)
’
o
R
2
and
V
s,
in
[0,
T].
Chacon
and
Hou
show
ha
i
co
is
app oxima ed
by
a
piecewise
polynomial
unc ion
on
polygons,
hen
he
co esponding
eloci y
gi en
by
(2)
may
be
compu ed
analy ically.
This,
oge he
wi h
(5),
sugges s
ha
we
app oxima e
he
o ici y
by
ini e
elemen s
on
a
mo ing
g id
whose
nodes
desc ibe
s eamlines
o
he
low.
The
algo i hm
o
Chacon
and
Hou
is
based
upon
his
idea.
Al hough
i s
desc ip ion
needs
a he
complex
no a ion,
we
shall
p o ide
i
as
needed
h oughou
his
pape .
Conside
a
iangula ion
Th
o
R
2
wi h
iangles
{727}i
u
and
nodes
{Olj}j6
u.
We
shall
assume
ha
h
deno es
he
leng h
o
he
longes
side
o
all
iangles
o
Th.
Deno e
by
Vh
he
space
o
con inuous
piecewise
a ine
ini e
elemen s
on
iangula ion
Th,
de ined
by
(6)
Vh
Vh
C
0
(RZ)
hl
is
a ine
o
all
iangles
6
Th
}.
A
unc ion
Vh
Vh
is
uniquely
de e mined
by
he
alues
Vh
(o j)
’i
iV.
1428
IBRAHIM
BLESS
RANERO
AND
TOM/S
CHACON
REBOLLO
Assume
ha
we
know
an
app oxima ion
Xj
o
he
poin
X
( n;
0,
o j)
o
each
j
6
N.
De ine
he
app oxima e
unc ion
X
6
Vh
o
he
low
map
X
( ;
0,
.),
espec i ely,
in
such
a
way
ha
i
e i ies
^n
’j
6N.
(c
s)
x
^n
Then,
7
{ i
( /)
’4
/
Th}
de ines
a
iangula ion
o
R
e,
p o ided
he
iangles
do
no
o e lap.
I
his
is
he
case,
we
also
de ine
he
piecewise
a ine
ini e
elemen
space
Q
on
iangula ion
7
in
he
same
way
ha
Vh
is
de ined
on
Th
in
(6).
We
shall
deno e
by/3
(x;
{coj
})
he
canonical
in e pola ion
ope a o
on
P,
de ined
by
We
shall
also
deno e
by
di)j
}jEN
he
canonical
base
o
V
h
Each
(Ioj
is
uniquely
de ined
by
i j
k,
j(O k)--
0
i jCk.
We
a e
now
eady
o
s a e
he
Chacon-Hou
algo i hm.
ALGORITHM
A.
Suppose
ha
coo
is
o
compac
suppo .
Deno e
wj
co0(o j).
1.
Ini ializa ion
(i)
T iangula ion;
(ii)
Vo ici y;
(iii)
Veloci y;
^0
X
=o j
and
T=Th.
2.
Time
i e a ion
(i)
Upda e
Lag angian
mesh
poin s
by
he
second-o de
Adams-Bash o h
me hod,
^n+l
"n
A
[
^n(])
^n-l( y-1
]
Xj
Xj
--
-
3
l,
h
--l,
h
(ii)
Cons uc
a
piecewise
linea
app oxima ion
o
X
( ;
0,
.),
^n+l
j
n+l
Z
Xj
jEN
(iii)
Upda e
o ici y;
()+1
(X)
/3+1
(X,
{O)j
}).
(i )
Upda e
eloci y;
^n+l
+1
u
h
=K*d
Algo i hm
A
may
be
iewed
as
a
o ex-blob
me hod
wi h
a
cu o
unc ion
a ying
in
space
and
ime
and
nono e lapping
smoo hing
pa ame e
3
h.
Thus,
Algo i hm
A
sha es
wi h
LONG
TIME
ACCURATE
VORTEX
METHOD
1429
o ex
me hods
he
p ope y
o
being
nondissipa i e.
Fu he mo e,
i
is
con e gen
wi hou
o e lapping
he
smoo hing
"blobs."
Chacon
and
Hou
p o e
ha
Algo i hm
A
is
con e gen
in
he
uni o m
no m
wi h
second-
o de
accu acy
unde
some
egula i y
p ope ies
o
he
amily
o
ini ial
iangula ions.
Accu-
acy
o
o de
highe
han
one
appea s
o
be
c ucial
o
ensu e
he
s abili y
o
he
me hod.
Some
nume ical
es s
show
ha
his
me hod
is
accu a e
only
o
ela i ely
sho
ime
in e als,
much
as
a e
o ex-blob
me hods
(see
Pe lman
[20]).
Howe e ,
a
la e
imes
he
iangula ions
become
degene a ed
and
accu acy
is
p og essi ely
los
in
a
sho
ime.
Thus,
eg idding
echniques
a e
needed
o
inc ease
he
in e al
o
accu acy
o
he
me hod.
Chacon
and
Hou
p o e
ha
he
me hod
is
con e gen
o
longe
ime
in e als
i
some
speci ic
eg idding
echniques
a e
in oduced.
This
occu s
in
pa icula
i
he
g id
nodes
a e
eg ouped
o
o m
new
iangula ions,
so
ha
hei
egula i y
is
conse ed.
3.
Fas
compu a ion
o
disc e e
eloci ies.
The
me hod
o
compu a ion
o
disc e e
e-
loci ies
in oduced
by
Chacon
and
Hou
equi es
compu a ions
o
o de
O(Ne),
whe e
N
is
he
numbe
o
g id
poin s
in
he
suppo
o
w
h."
o
This
makes
he
algo i hm
slow
e en
o
mode -
a ely
la ge
g id
sizes.
In
his
sec ion
we
de elop
an
adap a ion
o
he
algo i hm
in oduced
in
G eenga d
and
Rokhlin
[12]
(c .
also
G eenga d
[13])
ha
compu es
an
app oxima ion
o
he
disc e e
eloci ies
by
means
o
Taylo
expansions.
This
educes
he
compu a ional
complexi y
o
O
(N
log
e
N).
We
omi
mos
o
he
p oo s
o
he
esul s
p esen ed
in
his
sec ion,
as
hey
a e
adap a ions
o
he
co esponding
ones
gi en
by
G eenga d
and
Rokhlin.
Le
Th
be
a
ian-
gula ion
o
R
e
wi h
nodes
{o j
}jsN.
We
assume
a
ce ain
uni o mi y
in
he
spa ial
dis ibu ion
o
he
g id
nodes
o
Th.
This
is
needed
o
ensu e
he
con e gence
o
he
in e pola ion
by
ini e
elemen s
(c .
Cia le
11
]).
We
a e
gi en
a
o ici y
oh
Vh,
whe e
Vh
deno es
he
space
o
piecewise
a ine
elemen s
on
Th
de ined
in
(6).
Mo eo e ,
we
assume
ha
h
has
compac
suppo .
3.1.
T unca ed
expansions.
Ou
i s
goal
is
o
ob ain
a
unca ed
Lau en
expansion
ha
app oxima es
he
a - ield
eloci y
induced
by
he
o ici y
suppo ed
by
a
gi en
subse
Q
o
R
2.
This
o ici y
is
gi en
by
(7)
&Q
(i
(I)i.
o i
a
De ine
he
se
Qh--
{x
R2
/
d(x’
Q)--
in
lx-
yl
<-h
}
Then,
supp
()Q
may
include
Q,
bu
in
any
case
supp
OQ
C
Qh.
Deno e
by/Q
(b/l,
b/2)
he
eloci y
induced
by
&O"
K(x
x’)
OQ(X’)
IQ(X)
Jsu
PP Q
Then,
H
ul
iu2
is
a
unc ion
o
complex
a iables,
analy ic
in
C
supp
&a,
as
he
eal
and
imagina y
pa s
o
H
sa is y
he
Cauchy-Riemann
condi ions
in
C
supp
&a.
Mo eo e ,
(8)
/(Z)
pp Q
Z-
Z
Q(x )dx ’
whe e
Z
Xl
"3 "
ix2,
Z
X
X
-q-
lX2,
(X l
X2).

1430
IBRAHIM
BLESS
RANERO
AND
TOMAS
CHACON
REBOLLO
This
allows
us
o
expand//in
a
Lau en
se ies
as
ollows.
THEOREM
3.1.
Assume
supp
_Oa
(
D(zo,
),
whe e
D(zo,
)
deno es
he
complex
disc
o
cen e
zo
and
adius
>
O.
Then,
(9)
5/(z)
(Z
Z0)
k+l
V
z
C
D(Zo,
),
k=0
whe e
(10)
1
su
(z’
zo)koQ(X
’)
dx’.
ak
/
PP Q
Mo eo e ,
gi en
p
>
1,
le
us
deno e
by
L p
(z)
he
p- e m
unca ed
expansion
(9).
Then,
he
ollowing
e o
es ima e
holds:
(11)
I(z)
-p(z)l
_<
Iz
zol-
Iz
z01
i
Iz
z0l
>
,
whe e
1
Ial.
(12)
A
---
PP a
The
coe icien s
a
gi en
by
(11)
may
be
compu ed
analy ically
by
means
o
a
complex
e sion
o
G een’s
heo em.
Indeed,
as
supp
(Q
is
a
eunion
o
iangles
o
Th,
o
each
such
iangle
he e
exis
b ,
c ,
and
d
such
ha
1_
1
()QI
(371’
372)
b
xl
-b
c
x2
q-
d
P
Z
+
P
q-
d ,
whe e
p
=
b
+
c .
Thus,
1
C
supp
5
Q
z
(z
zo)
g
z
dz
+
-
p
(z
zo)
dz
+
d
(z
zo)
k
dz
The
men ioned
complex
e sion
o
G een’s
o mula
is
used
o
compu e
he
in eg als
in
(3.1).
I
is
s a ed
as
ollows.
LEMMA
3.2.
Le
B
be
a
bounded
measu able
subse
o R
2
wi h
Lipschi z
bounda y
0
B.
Le
and
g
be
wo
unc ions
o
complex
alues
de ined
and
analy ic
in
some
open
se
con aining
B
U
0
B.
Then,
1
(z)
g(z)
dz.
’z
gz
dz
Le
us
now
conside
a
iangle
Th
included
in
supp
()Q.
By
aking
(z)
z,
e,(z)
(z
zo)
z,
we
ob ain
1
)
(z
zo)
z
dz
z
(z
zo
dz.
LONG
TIME
ACCURATE
VORTEX
METHOD
1431
Also,
by
aking
Z
2
(z)
g(z)
(z
Zo)
2’
o2
(z
zo)
z
z
(z
zo)
clz.
A
simila
choice
allows
us
o
ind
he
las
e m
in
(3.1).
Finally,
he
compu a ion
o
he
coe icien s
a
educes
o
ha
o
polynomials
on
segmen s
o
s aigh
lines,
which
is
ob ained
analy ically.
Ou
pu pose
now
is
o
shi
he
cen e s
o
he
unca ed
Lau en
expansion
ob ained
abo e
and
o
con e
hese
expansions
in o
Taylo
expansions.
We
do
his
in
he
nex
h ee
lemmas.
LEMMA
3.3.
Assume
supp
Oa
Q
D(zo,
).
Le
zl
C
be
such
ha
Iz0
zi[
>
.
Then,
bk
(13)
(z)
=
(Z-
Zl)
k+l
k=0
whe e
Y
z
C
D(zl,
Izl
z0l
+
),
(14)
b
Z
a (z
ZO)
k-l.
/=0
Mo eo e ,
i
we
deno e
by
ld
he
p- e m
unca ed
expansion
(13),
he
ollowing
e o
es ima e
holds:
A
[,Zl-ZOl+ ]
p+I
(15)
I/d(z)
-/d(z)l
_<
Iz
zl-
(]Zl
zol
+
)
Iz
Zol
whe e
The
ans o ma ion
o
Lau en
expansions
in o
Taylo
expansions
is
made
as
ollows.
LEMMA
3.4.
Assume
supp
&a
C
D(zo,
).
Le
z
C
be
such
ha
IZl
z0[
>
(c
+
1)
(16)
(z)
=
/k(Z-
Zl)
k
/Z
D(Zl,
),
k=0
whe e
(k+l)
a
(17)
(-1)
(z
zo)
+
o=
(zl
zo)
l"
Mo eo e ,
gi en
an
in ege
numbe
q
>
1,
de ine
he
unca ed
expansion
p
(18)
Upq(Z)
/k
(Z
Zl)
k,
k=O
whe e
he
coe icien s
,
a e
de ined
by
k
al
(19)
=
(-1)
(z-
z0)
k+l
(Zl
-zo)
l"
A
2
ppQ
o
some
c
>
1.
Then,
1432
IBRAHIM
BLESS
RANERO
AND
TOMAS
CHAC 3N
REBOLLO
Then,
he
ollowing
e o
es ima e
holds:
(20)
[(z)
LCpq(Z)]
c-1 c-1
+1
whe e
1
u
IQI,
A
PPQ
Z
G
D(Zl,
),
Lemma
3.4
p o ides
an
e o
es ima e
o
a
unca ed
expansion
l/[pq
ha
is
de ined
only
wi h
a
ini e
numbe
o
da a:
he
p
+
q
coe icien s
a
l,
a2
ap+q
u nished
by
ei he
Theo em
3.1
o
Lemma
3.3.
The
ansla ion
o
he
cen e s
o
he
unca ed
Taylo
expansions
is
gi en
as
ollows.
LEMMA
3.5.
Gi en
he
complex
numbe s
Zl,
Z2;
/91,/91
pp,
he
ollowing
holds:
p
p
(21)
&
(z
z)
/Sk
(Z
Z2)
,
k=0
l=0
whe e
(22)
p
/k
/gk
(Z2
Zl)
k-l.
k=l
Obse e
ha
as
o mula
(21)
is
exac ,
he
e o
bound
(20)
is
s ill
ue
o
he
unca ed
Taylo
expansion
wi h
shi ed
cen e
i
z2
6
D(Zl,
)
and
z
D(z2,
]Zl
z21).
Also,
assume
ha
he
p
a e
he
coe icien s
o
he
Taylo
expansion
o /g
a ound
z.
Then,
in
gene al,
he/5l
a e
no
he
coe icien s
o
b/a ound
z2.
This
would
happen
only
i
he
sum
in
(22)
we e
in ini e.
3.2.
Desc ip ion
o
he
algo i hm.
To
desc ibe
o
some
ex en
he
algo i hm
o
as
compu a ion
o
he
disc e e
eloci y,
we
shall
need
some
speci ic
de ini ions.
We
assume
ha
he
suppo
o
he
o ici y
h
is
included
in
a
squa e
o
sides
o
leng h
H,
=
2’2
x
2’2
We
e e
o
his
squa e
as
he
compu a ional
domain.
We
subdi ide
he
compu a ional
domain
in o
a
amily
o
boxes
o
dec easing
size
which
will
be
linked
by
a
hie a chy
ela ion.
Le
N
be
he
numbe
o
nodes
Oem
in
supp
(Sh,
and
de ine
he
highes
le el
o
e inemen :
J
log
4
N.
Gi en
a
le el
o
e inemen
j
0,
1
J,
we
deno e
by
Q,
k
1,
2
4J
he
squa es
/-/
We
deno e
by
ob ained
by
spli ing
each
side
o
in o
2
j
subin e als
o
leng h
hj
2-7.
/Jk
he
cen e
o
box
Q.
We
assume
ha
each
box
Q
a
le el
j
is
a
Ca esian
p oduc
o
in e als
o
he
o m
Q
[a,
b[
x
[c,
d[
H
o
some
a
<
b
and
c
<
d
in
[-7,
["
Then,
he
boxes
a
le el
j
do
no
o e lap,
bu
hei
eunion
is
he
whole
compu a ional
box
LONG
TIME
ACCURATE
VORTEX
METHOD
1433
DEFINITION
3.6.
Gi en
d
>
O,
we
shall
say
ha
wo
boxes
Q
and
Q
J
o
he
same
le el
j
a e
d-sepa a ed
i
k
--i11
"
d
hj.
The
ac
ha
wo
boxes
a e
well
sepa a ed
allows
us
o
app oxima e
uni o mly
on
each
one
o
hem
he
eloci y
ield
induced
by
he
o ici y
suppo ed
by
he
o he
one
by
means
o
he
unca ed
Taylo
de elopmen
u nished
by
Lemmas
3.4
and
3.5.
To
s a e
his
esul
we
need
o
gi e
a
o mal
de ini ion
o
he
aspec
a io
o
a
iangula ion,
as
a
measu e
o
i s
egula i y.
The
aspec
a io
o
a
gi en
iangle
is
he
a io
be ween
he
diame e
o
he
smalles
ci cle
ha
can
ci cumsc ibe
he
iangle
and
ha
o
he
la ge
ci cle
ha
can
be
insc ibed
in
he
iangle.
The
aspec
a io
o
a
iangula ion
Th
is
he
la ges
o
all
aspec
a ios
o
all
iangles
o
Th.
LEMMA
3.7.
Gi en
a
box
Q
o
le el
j,
deno e
by
co
Vh
he
o ici y
suppo ed
by
QJ
(23)
O m
E
Q
Call
l
he
(p,
q)- e m
Taylo
expansion
associa ed
o
o9
de ined
by
(18).
De ine
Le
c
>
1
be
gi en.
Then,
he e
exis s
a
cons an
)
>
O.depending
only
on
he
a.spec
a io
o
iangula ion
Th
such
ha
i
d
(c
+
1)
and
box
QJ
is
d-sepa a ed
om
QJ,
hen
C-+-1
(24)
xEQ{max
lu
uJl
_<
B
hj
(c
1)2
+
c
+1
Il h
whe e
B
is
a
cons an
depending
only
on
he
aspec
a io
o
iangula ion
Th.
P oo .
As
h
is
he
longes
leng h
o
all
iangles
o
Th,
hen
(25)
supp
coJ
C
B( lJ,
j)
wi h
j
--
hj
+
h.
Also,
as
h
j
,
he e
exis
wo
posi i e
cons an s
and/z
depending
only
on
he
aspec
a io
o
T,
such
ha
h<hj< zh.
Thus,
1
1
j
<
----_
hj
+-hs
<_.
)hj,
/
2
As
he
boxes
a e
d-sepa a ed,
1 1
whe e
)
+
-’
d
l
ill>
d
h
j
>
.
I
we
ake
d
)
(c
+
1),
we
ha e
he
hypo heses
o
Lemma
3.4
wi h
zo
l
J
k’
Zl
[l
j.
Es ima e
(24)
ollows
immedia ely.
1440
IBRAHIM
BLESS
RANERO
AND
TOM,S
CHAC 3N
REBOLLO
FIG.
5.
Nume ical
con e gence
o de
in
eloci y
o
Algo i hm
A’,
applied
o
TC
1,
o
a
sho
ime
in e al.
The
lines
ma ked
by
and
+
symbols
co espond,
espec i ely,
o
hi
1/12,
h:
1/16
and
o
hi
1/16,
h2
1/20
o
compu e
he
nume ical
con e gence
o de .
The
co esponding
heo e ical
con e gence
o de
is
p
2
(line
ma ked
by
0
symbols).
FIG.
6.
Nume ical
con e gence
o de
in
eloci y
o
Algo i hm
A’,
applied
o
TC
1,
o
a
sho
ime
in e al.
The
lines
ma ked
by
and
+
symbols
co espond,
espec i ely,
o
h
/
12,
h2
/
16
and
o
h
/
16,
h2
1/20
o
compu e
he
nume ical
con e gence
o de .
The
co esponding
heo e ical
con e gence
o de
is
p
1.5
(line
ma ked
by
0
symbols).
In
Figs.
5
and
6
we
ep esen
he
beha iou
o
he
nume ical
con e gence
o de
in
eloci y
o
TC
1
o
sho
imes.
Figu e
5
co esponds
o
a
heo e ical
con e gence
o de
p
2,
ob ained
by
choosing
a
2
in
Algo i hm
A’.
Figu e
6
co esponds
o
p
1.5,
ob ained
by
choosing
c
1.5.
The
sha p
oscilla ions
obse ed
in
hese
cu es
occu
when
Deiaunay
eg idding
is
pe o med.
In
bo h
cases
he e
is
a
good
ag eemen
be ween
he
compu ed
con e gence
o de
and
he
heo e ical
one.
Fu he mo e,
he
cu es
co esponding
o
smalle
alues
o
h
a e
globally
close
o
he
heo e ical
con e gence
o de s.
No e
ha
in
all
cases
he
compu ed
o de s
oscilla e
a ound
he
heo e ical
alue.
I
is
in e es ing
o
obse e
ha
he
con e gence
o de
o
Algo i hm
A’
may
be
p ese
o
any
alue
p
6]
1,
2]
by
simply
choosing
he
pa ame e
c
p
a
he
beginning
o
he
un.
Howe e ,
aking
smalle
han
2
would
p oduce
a
was e
o
compu a ional
wo k,
as
he
compu a ional
complexi y
o
Algo i hm
A’
is
o
o de
N
log
2
N
o
any
alue
o
a.
In
wha
ollows
we
shall
always
ake
2.

LONG
TIME
ACCURATE
VORTEX
METHOD
1441
o s
FIG.
7.
Time
e olu ion
o
nume ical
con e gence
o de
in
o ici y
o
Algo i hm
A
,
applied
o
TC2,
o
he
ime
in e al
[0,
15].
The
lines
ma ked
by
+,
*,
and
0
symbols
co espond,
espec i ely,
o
hi
1/12,
h2
1/16;
hi
1/16,
h.
1/20;
and
o
hi
1/16,
h2
1/20
o
compu e
he
con e gence
o de .
A
good
ag eemen
wi h
he
heo e ical
p edic ion
p
2,
which
imp o es
o
smalle
alues
o
h,
is
obse ed
o
he
whole
ime
in e al.
3"0
1
2.5
E o s
FIG.
8.
Time
e olu ion
o
nume ical
con e gence
o de
in
eloci y
o
Algo i hm
A
,
applied
o
TC2,
o
he
ime
in e al
[0,
15].
The
lines
ma ked
by
+,
*,
and
0
symbols
co espond,
espec i ely,
o
hi
1/12,
h2
1/16;
hi
1/16,
h2
1/20;
and
o
hi
1/16,
h2
1/20
o
compu e
he
con e gence
o de .
A
goodag eemen wi h
he
heo e ical
p edic ion
p
2,
which
imp o es
o
smalle
alues
o
h,
is
obse ed
o
he
whole
ime
in e al.
Figu es
7
and
8
show
he
nume ical
con e gence
o de
in
o ici y
and
eloci y
o
TC2
du ing
a
ela i ely
long
in eg a ion
ime.
The
sha p
oscilla ions
o
he
o me
es
a e
s ill
obse ed.
Howe e ,
again
he
cu es
co esponding
o
smalle
alues
o
h
a e
close
o
he
heo e ical
o de
p
2.
No e
ha
he
nume ical
o de s
in
eloci y
a e
close
o
p
2
han
hose
in
o ici y.
This
is
p obably
due
o
he
highe
egula i y
o
he
eloci y
ield.
No e
also
ha
he
ag eemen
holds
e en
o
long
imes.
5.3.
Beha iou
o
long
in eg a ion
imes.
Ou
hi d
se
o
nume ical
expe imen s
deals
wi h
he
analysis
o
he
long
ime
beha iou
o
e o s
due
o
Algo i hm
A’.
A
i s ,
we
es ed
he
e ec
o
in oducing
Delaunay
eg idding
in
Algo i hm
A’.
In
Fig.
9
we
ep esen
he
ime
e olu ion
o
he
pe cen
ela i e
e o s
in
eloci y
co esponding
1442
IBRAHIM
BLESS
RANERO
AND
TOM/S
CHACON
REBOLLO
o
E o s
FIG.
9.
Time
e olu ion
o
ela i e
e o s
in
o ici y
o
Algo i hm
A
wi h
and
wi hou
using
Delaunay
eg idding
(ma ked
by
and
+
symbols,
espec i ely).
The
second
one
g ows
exponen ially
while
he
i s
one
keeps
close
o
he
ini ial
e o .
o
Algo i hm
A
wi hou
eg idding
and
o
Algo i hm
A’
using
Delaunay
eg idding,
applied
o
TC
wi h
h
/
12.
We
made
a
un
in
he
ime
in e al
[0,
50],
which
appea s
as
qui e
a
long
ime
o
he
es
case
conside ed.
Indeed,
in
ha
in e al
he
as es
poin s
in
supp
09
comple ed
mo e
han
h ee
o a ions,
while
he
slowes
ones
did
no
comple e
hal
a
o a ion.
No e
ha
Algo i hm
A
is
s ill
de ined
when
he
iangula ion
TT,
becomes
degene a ed.
Indeed,
in
his
case
i
is
s ill
possible
o
compu e
di ec ly
he
exac
eloci y
T.
We
may
obse e
ha
in
his
case
he
e o
g ows
exponen ially
un il
a emp ing
ela i e
alues
o
mo e
han
50%
a
_
50.
When
Delaunay
eg idding
is
used,
he e
is
a
all
o
e o
each
ime
i
is
e ec i ely
pe o med.
This
is
due
o
he
diminishing
o
he
local
g id
size
ha
ende s
he
linea
in e pola ions
on
each
iangle
mo e
accu a e.
A e
his,
he e
is
a
slow
exponen ial
inc ease
o
e o s,
which
alls
again
he
nex
ime
Delaunay
eg idding
is
used.
No e
ha
eg idding
is
no
needed
e y
o en.
The
e o
a
50
is
app oxima ely
only
wo
imes
he
ini ial
one.
Figu es
10
and
11
also
show
he
beha iou
o
e o s
in
eloci y
and
o ici y
co esponding
o
TC1
and
TC2,
espec i ely,
wi h
h
1!12,
du ing
he
ime
in e al
[0,
100].
This
is
a
e y
long
ime
in e al
o
bo h
cases,
as
a
ime
100
he
uni
ci cle
has
been
d ama ically
de o med
by
bo h
lows.
The
cu es
p esen
sha p
oscilla ions
due
no
only
o
he
use
o
Delaunay
eg idding,
bu
also
o
he
low
smoo hness
o
he
disc e e/-no m
used.
Howe e ,
we
ema k
a
i s
ha
in
bo h
cases
he
e o s
in
eloci y
and
o ici y
emain
almos
cons an .
Also,
in
bo h
cases
he
e o s
in
eloci y
a e
subs an ially
smalle
han
hose
in
o ici y.
Again,
his
is
e y
p obably
due
o
he
highe
smoo hness
o
he
eloci y
ields.
No e
also
ha
al hough
in
TC2
he
o ici y
is
no
smoo h
enough
o
ensu e
he
con e gence
o
Algo i hm
A’,
in
p ac ice
second-o de
con e gence
is
a emp ed.
A
possible
eason
o
his
ac
is
ha
he
singula i ies
o
he
o ici y
lie
on
he
cu e
1,
while
we
sol e
Eule
equa ions
only
inside
he
uni
ci cle.
Also,
he
ac
ha
he
e o s
co esponding
o
TC2
a e
close
o
hose
co esponding
o
TC
1
is
p obably
due
o
he
adial
dis ibu ion
o
he
nodes
in
he
iangula ion
used.
Figu e
12
ep esen s
he
iangula ion
o
TC1
a
ime
99,
he
las
ime
Delaunay
eg idding
is
used.
Obse e
he
good
quali y
o
he
g id,
which
sugges s
ha
ou un
could
con inue
o
longe
imes
wi h
simila
e o
le els.
Globally,
hese
es s
show
ha
Algo i hm
A’
wi h
he
use
o
Delaunay
eg idding
is
s able
and
accu a e,
wi h
second-o de
accu acy,
e en
o
e y
long
in eg a ion
imes.
LONG
TIME
ACCURATE
VORTEX
METHOD
1443
E o s
2.5
2.0
Z.
5
FIG.
10.
Time
e olu ion
o
e o s
in
eloci y
(line
ma ked
by
+
symbols)
and
o ici y
(line
ma ked
by
symbols)
o
Algo i hm
A’,
applied
o
TC1
wi h
h
1/12,
in
he
ime
in e al
[0,
100].
Bo h
e o s
emain
close
o
he
ini ial
alues
o
he
whole
ime
in e al.
E o s
2.5
2.0
0
9
18
27
36
45
54
63
72
81
90
99
FIG.
11.
Time
e olu ion
o
e o s
in
eloci y
(line
ma ked
by
+
symbols)
and
o ici y
(line
ma ked
by
symbols)
o
Algo i hm
A
,
applied
o
TC2
wi h
h
1/12,
in
he
ime
in e al
[0,
100].
Bo h
e o s
emain
close
o
he
ini ial
alues
o
he
whole
ime
in e al.
5.4.
Compa ison
o
a
desingula ized
o ex
me hod.
Ou
nex
expe imen
is
o
com-
pa e
he
pe o mances
o
a
o ex
me hod
on
a
ixed
uni o m
g id
wi h
hose
o
Algo i hm
A’
using
Delaunay
eg idding.
As
he
o ex
me hod
we
ha e
used
he
desingula ized
poin
o ex
me hod
(DPVM)
in oduced
in
Hou
[15].
To
desc ibe
i ,
le
us
conside
a
uni o m
g id
o
size
h
o
R
2
wi h
nodes
{ lj
}j
EN.
Deno e
coj
COO
(/j).
Then,
DPVM
compu es
he
disc e e
eloci y
a
poin /3
and
ime
by
(43)
h( lk,
)
g(k
l)
(COl
(-Ok)
COk
I
g(
k
y)
dy,
/l
Esupp
wo
J 2h
( )
whe e
]h
( )
is
a
polygonal
app oxima ion
o
supp
co(.,
).
1444
IBRAHIM
BLESS
RANERO
AND
TOM,/ S
CHACON
REBOLLO
FIG.
12.
T iangula ion
a
ime
99
co esponding
o
Algo i hm
A
,
applied
o
TC2
wi h
h
1/12,
in
he
ime
in e al
[0,
100].
This
is
a
s able
modi ica ion
o
he
poin
o ex
me hod,
uni o mly
con e gen
wi h
second-o de
accu acy.
Because
o
ha ,
i
seems
o
be
a
good
me hod
o
compa e
wi h
ou s.
In
ou
expe imen s
we
ha e
un
bo h
algo i hms
o
g ids
o
size
h
1/12
and
h
1/16.
We
always
ake
he
uni
ci cle
o
be
he
se
2h( ).
This
allows
us
o
compu e
exac ly
he
in eg al
exp ession
in
(43).
Also,
we
sol ed
he
equa ion
o
cha ac e is ics
o
he
DPVM
wi h
he
Adams-Bash o h
second-o de
scheme,
jus
as
in
Algo i hm
A’.
In
ou
es s,
i
no
eg idding
echniques
a e
in oduced,
he
DPVM
p oduces
a
la ge
inc ease
o
e o s
in
a
ela i e
ime
in e al.
Fo
ins ance,
o
TC
he
ela i e
e o s
in
eloci y
ake
alues
o
app oxima ely
80%
by
ime
40.
As
we
poin ed
ou
in
he
In oduc ion,
some
con enien
eg idding
echnique
is
needed
o
ob ain
accu a e
solu ions
o
long
in eg a ion
imes.
Beale
and
Majda
in oduced
in
1985
a
simple,
bu
e icien ,
eg idding
echnique
in
he
con ex
o
a
o ex-blob
me hod
(VBM).
VBMs
a e
based
upon
he
disc e iza ion
o
o ici y
as
a
sum
o
smoo h
unc ions
wi h
small
suppo s,
called
blobs.
Gi en
a
smoo h
cu o
unc ion
q
(i.e.,
an
app oxima ion
o
he
Di ac
del a
a
he
o igin),
he
o ici y
a
a
ixed
ime
is
app oxima ed
by
(44)
co(x,
)
coh(X,
)
qa(x
Xj( ))
COj
h
2,
J
whe e
1
%(x)
Wi h
a
disc e iza ion
o
he
kind
o
(44),
i
is
possible
o
compu e
he
o ici y
a
any
p esc ibed
poin .
The
eg idding
echnique
o
Beale
and
Majda
consis s
o
ein e pola ing
he
o ici y
a
he
nodes
o
a
uni o m
g id,
whene e
he
cu en
local
g id
size
is
la ge
enough.
Howe e ,
as
epo ed
by
Beale
and
Majda,
he
g id
size
o
he
ein e pola ing
g id
should
dec ease
p og essi ely
o
main ain
easonable
e o
le els.
In
DPVM
he
o ici y
is
disc e ized
as
a
sum
o
Di ac
masses:
(45)
co(x,
)
--
cob(X,
)
Z
6(X
Xj( ))
coj
h
2.
J
LONG
TIME
ACCURATE
VORTEX
METHOD
1445
4
0 9
18
27
36
45
54
69
72
81
90
99
FIG.
13.
Compa ison
o
e o s
in
eloci y
be ween
he
DPVM
(line
ma ked
by
+
symbols)
and
Algo i hm
A
(line
ma ked
by
symbols),
applied
o
TC1,
in
he
ime
in e al
[0,
100].
A
p og essi e
inc ease
is
obse ed
in
he
i s
one,
while
he
second
emains
almos
cons an .
Consequen ly,
he
eg idding
echnique
o
Beale
and
Majda
canno
be
di ec ly
applied
he e.
Howe e ,
i
is
possible
o
use
his
echnique
a e
app oxima ing
d
by
a
sum
o
blobs
as
in
(44).
In
p ac ice,
we
ha e
used
a
ou h-o de
cu o
unc ion
:
*8(x)=-
2exp
--
-
This
cu o
unc ion
was
also
in oduced
by
Beale
and
Majda
in
1985.
Fo
smoo h
unc ions
,
he
e o
a,
is
o
o de
64.
We
ha e
aken
6
o
o de
h,
so
his
accu acy
seems
o
be
enough,
as
he
DPVM
is
o
o de
h
2.
The
eg idding
s a egy
ha
we
ha e
used
consis s
o
ein e pola ing
he
o ici y
when
he
smalles
angle
o
he
de o med
g id
is
smalle
han
a
p ese
limi
alue.
Each
eg idding
has
been
se
o
p oduce
an
inc ease
in
he
amoun
o
he
g id
poin s
o
app oxima ely
15%.
Figu es
13
and
14
compa e
he
beha iou
o
e o s
due
o
he
DPVM
and
o
he
ini e
elemen
o ex
me hod
(FEVM)
o
Algo i hm
A’.
We
ep esen
he
e o s
in
eloci y
and
ajec o ies
co esponding
o
TC1
du ing
he
ime
in e al
[0,
100]
wi h
ini ial
g id
size
h
1/16.
Sha p
a ia ions
o
e o s
co esponding
o
he
DPVM
a e
obse ed,
p obably
due
o
he
addi ional
e o
in oduced
in
he
ein e pola ion
associa ed
o
he
eg idding
s eps.
The
e o s
co esponding
o
he
DPVM
inc ease
as e
han
hose
co esponding
o
he
FEVM.
By
ime
0,
he
e o s
co esponding
o
bo h
me hods
ake
e y
simila
alues.
By
ime
100,
he
la e
e o s
a e
nea ly
200
imes
smalle
han
he
o me
ones
in
eloci y
and
nea ly
20
imes
smalle
in
ajec o ies.
This
di e en
g ow h
a e
is
p obably
a
consequence
o
he
in oduc ion
o
nume ical
di usion
in
he
eg idding
s eps.
We
may
conclude
ha
he
FEVM
sol es
mo e
accu a ely
ou
TC
1
o
long
imes,
wi hou
in oducing
nume ical
di usion.
We
should
poin
ou
ha
he
compu a ional
wo k
needed
by
one
ime-s ep
wi h
he
FEVM
is
nea ly
100
imes
bigge
han
he
one
needed
by
one
ime-
s ep
wi h
he
DPVM.
Howe e ,
his
wo k
emains
cons an
in
ime
o
he
FEVM,
while
ha

1446
IBRAHIM
BLESS
RANERO
AND
TOM/S
CHAC(3N
REBOLLO
2.5
-0.
E o s
1(3
27
3(3
45
54
63
72
131
90 99
FIG.
14.
Compa ison
o
e o s
in
ajec o ies
o
g id
poin s
be ween
he
DPVM
(line
ma ked
by
+
symbols)
and
Algo i hm
A
(line
ma ked
by
symbols),
applied
o
TC1,
in
he
ime
in e al
[0,
100].
needed
by
he
DPVM
inc eases
each
ime
eg idding
is
pe o med.
Thus,
o
long
enough
ime
in e als,
bo h
compu a ional
e o s
will
be
o
he
same
o de .
5.5.
Compa ison
o
high-o de
VBMs.
We
inally
compa ed
he
FEVM
wi h
he
VBM
wi h
cu o
unc ions
o
o de s
m
4
and
m
6.
Speci ically,
we
used
hose
in oduced
by
Beale
and
Majda,
co esponding
o
p:4,
(x)=-
2exp
-
-
exp
(-
2@2)]
p--6,
1
2
1
2
J6)(x)
-
I
exp
(----)-
exp
(--2@2)+
-
exp
(---)].
To
ensu e
he
con e gence
o
he
me hod,
we
ook
he
blob
size
o
be
6
h
q,
wi h
0
<
q
<
1.
Thus,
o
smoo h
enough
ini ial
o ici y,
he
con e gence
o de
o
he
me hod
is
p=mq.
In
p ac ice,
we
ook
q
0.95
in
all
ou
expe imen s,
as
his
alue
seems
o
be
quasiop-
imal,
as
epo ed
by
Pe lman.
The
s eamline
equa ion
has
been
sol ed
wi h
a
ou h-o de
Runge-Ku a
me hod
wi h
e y
small
ime-s ep.
We
ha e
es ed
ou
code
o
TC1
a
1.
Ou
es ima ions
o
compu ed
con e gence
o de s
a e
gi en
in
Table
1.
They
a e
in
e y
good
ag eemen
wi h
hose
epo ed
by
Pe lman.
We
ha e
used
he
eg idding
echnique
desc ibed
in
he
p eceding
subsec ion,
wi h
some
mino
modi ica ions.
Indeed,
many
possible
c i e ia
which
can
be
used
o
apply
eg idding
a e
equi alen
in
p ac ice
o
he
o a ing
s eady
solu ions
we
a e
conside ing.
Ei he
eg idding
when
he
smalles
angle
o
he
mesh
is
smalle
han
a
gi en
ole ance,
o
when
he
cu en
g id
size
is
long
enough,
is
equi alen
o
eg idding
a
ce ain
ixed
numbe
o
ime-s eps.
In
any
LONG
TIME
ACCURATE
VORTEX
METHOD
1447
TABLE
Con e gence
o de
o
he
VBM
o
TC1,
es ima ed
a
ime
1,
used
o
compa e
o
he
FEVM.
Values
o
m
and
h
h
0.2
h
0.1
h
0.05
Theo e ical
o de s
m
4
2.53
3.32 3.60
3.80
m
6
2.78
4.44
5.16
5.70
O,B
0.7
IME
I
FIG.
15.
Compa ison
o
e o s
in
eloci y
be ween
Algo i hm
A’
(line
ma ked
by
0
symbols)
and
he
VBM
wi h
m
4
(line
ma ked
by
+
symbols)
and
m
6
(line
ma ked
by
symbols) o
TC1
and
h
1/16
in
he
ime
in e al
[0,
100].
case,
he
ac ual
ole ance
alue
mus
be
uned
wi h
ca e
o
a oid
an
excessi e
inc ease
in
e o s.
I
eg idding
is
applied
oo
o en,
we
shall
p og essi ely
in oduce
high
le els
o
nume ical
di usion,
bu
i
he
g id
is
excessi ely
dis o ed
when
eg idding,
hen
he
accumula ed
e o s
will
p oduce
an
un eco e able
loss
o
accu acy.
In
Figs.
15
and
16
we
ep esen
he
ela i e
e o s
in
eloci y
o
he
FEVM
and
o
he
VBM
wi hm
4
andm
6,
co esponding
o
TC1
wi h
h
1/16
and
TC2
wi h
h
1/12.
We
may
obse e
ha
eg idding
is
applied
in
all
cases
an
almos
cons an
numbe
o
ime-s eps.
In
all
cases
e o s
a e
kep
almos
cons an
o
a
sho
ime
in e al
whene e
eg idding
is
applied
and
expe ience
a
as
inc ease
when
he
g id
becomes
p og essi ely
dis o ed.
Fo
m
6,
his
inc ease
is
e y
as ,
and
his
is
p obably
he
eason
why
eg idding
p oduces
a
dec ease
in
e o s.
Fo
m
4,
he
e o s
do
no
g ow
as
as ,
and
eg idding
p oduces
an
inc ease
in
hem.
Also,
almos
linea
g ow h
a es
o
e o s
a e
obse ed
in
all
cases.
These
a es
a e
smalle
o
m
6
han
o
m
4.
Fo
he
FEVM,
he
g ow h
o
e o s
as
he
g id
is
dis o ed
is
he
as es
o
all
cases
conside ed.
Thus,
he
loss
o
quali y
o
he
g id
mo e
d ama ically
a ec s
he
accu acy
o
he
FEVM
han
ha
o
he
VBM.
Howe e ,
applying
Delaunay
eg idding
in
he
FEVM
diminishes
he
e o s
o
alues
close
o
he
ini ial
ones,
coun e balancing
almos
comple ely
he
o me
inc ease.
In
TC
1,
which
co esponds
o
a
smoo h
solu ion,
he
FEVM
p esen s
a
be e
pe o mance
a
ime
100
han
he
VBM
wi h
m
4,
while
he
VBM
wi h
m
6
yields
a
highe
accu acy
han
he
FEVM.
Howe e ,
in
TC2,
which
co esponds
o
a
less
smoo h
o ici y,
he
pe o mance
o
he
FEVM
a
100
imp o es
ha
o
he
VBM
wi h
m
4
and
also
ha
o
he
VBM
wi h
m
6.
We
should
also
ema k
ha
he
g ow h
a es
o
e o s
o
he
BVM
a e
1448
IBRAHIM
BLESS
RANERO
AND
TOM/S
CHACON
REBOLLO
(VEL.),_
H=/!
4050.
70,
100.
FIG.
16.
"Compa ison
o
e o s
in
eloci y
be ween
Algo i hm
A’
(line
ma ked
by
0
symbols)
and
he
VBM
(line
ma ked
by
+
symbols)
wi h
m
4
and
m
6
(line
ma ked
by
symbols) o
TC2
and
h
1/12
in
he
ime
in e al
[0,
100].
in
all
cases
la ge
han
hose
co esponding
o
he
FEVM.
Thus,
he
compa ison
is
e y
likely
o
be
e en
mo e
a ou able
o
he
FEVM
o
la e
in eg a ion
imes.
Finally,
we
mus
say
ha
we
may
no
expec
o
sol e
wo-dimensional
Eule
equa ions
wi h
any
ini ial
condi ion,
simply
by
using
Algo i hm
A’
combined
wi h
Delaunay
eg idding.
I
seems
clea
ha ,
in
gene al,
he
o he
eg idding
ules
ha
we
men ioned
in
2
a e
needed
o
ob ain
accu a e
esul s.
The
esul s
p esen ed
in
his
pape
mus
be
unde s ood
in
he
sense
ha
ou
FEVM,
due
o
i s
geome ical
adap abili y,
imp o es
he
accu acy
o
classical
o ex
me hods,
wi hou
in oducing
nume ical
di usion.
Appendix:
P oo
o
Theo em
4.1.
P oo .
Ou
p oo
is
an
adap a ion
o
he
con e gence
p oo
o
Algo i hm
A
gi en
by
Chacon
and
Hou.
The
essen ials
o
he
p oo
a e
as
ollows.
Le
us
de ine
T*
max
.
"0
<
n
<
T,
max
IlX
311,h
<
h
I+p
whe ep
[min(2
c )-l]
0<k<n
We
p o e
ha
he e
exis s
a
sepa a ion
pa ame e
d,
depending
only
on
Coo,
T,
c,
and
,
such
ha
es ima es
(33)
hold
in
he
ime
in e al
[0,
T*].
Then,
we
conclude
ha
he e
exis
wo
posi i e
numbe s
h
T-
and
AT-
such
ha
i
0
<
h
<
h
T-
and
0
<
A
<
AT-,
hen
he e
mus
be
T*>T.
The
main
inno a ion
in
ou
analysis
is
ha
we
ob ain
uni o m-in- ime
es ima es
o
he
e o
in
he
compu a ion
o
disc e e
eloci ies
by
Algo i hm
B.
As
we
shall
see,
his
essen ially
happens
because
ou
algo i hm
p ese es
he
uni o m
no m
o
he
disc e e
solu ion.
Indeed,
i
0
_<
,
_<
T*,
ollowing
Chacon
and
Hou
we
s a e
ha
i
h
is
small
enough,
hen
all
iangula ions
{7}0<_ ,<_,
a e
nondegene a ed.
Mo eo e ,
all
aspec
a ios
o
hese
iangula ions
a e
uni o mly
bounded
om
abo e
by
a
cons an
}/
independen
o
h.
Le
us
de ine
he
sepa a ion
pa ame e
d
)L
(1
+
c),
whe e
i
z
is
he
pa ame e
associa ed
o
?’
gi en
by
Theo em
3.11.
No e
ha
d7
depends
only
on
coo,
T,
c,
and
he
aspec
a io
o
he
ini ial
iangula ions.
LONG
TIME
ACCURATE
VORTEX
METHOD
1449
(46)
and
We
p o e
now
ha
he e
exis
wo
posi i e
cons an s
and
C
such
ha
supp
^n
o)
h
C
B(0,
),
0
<
n
<
T*,
(47)
Indeed,
assume
^n
.n.
h
T*.
max
I[ i--(K.(Oh)](
j)l
<C
0< n
<
O<j<M
supp
b-I
C
B(0,
n-1),
supp
^n
(-o
h
C
B(O,
n)
o
some
posi i e
numbe s
n-1
<
n.
Le
us
deno e
by
he
Euclidean
no m
on
R
2
and
by
II
he
uni o m
no m
on
R
2.
Theo em
3.11
yields
(48)
<
C1
n
^n
h
^n
%11
- -
Ig(]
Y)I
Coh(Y)dY
_<
C2
n
Ilco011
JB
(O, .)
No e
ha
he
las
inequali y
he e
ollows
because
Algo i hm
A
conse es
in
ime
he
uni o m
no m
o
he
disc e e
o ici y.
Consequen ly,
IXj^n+l
IXjl
-A
3
[u
h^n(Jy)[
_+_
[ n-l(j]-l)[
_<
IXjl
+
C3
A ,
j--1
M.
Thus,
n
<
o
exp(C3
T),
0
<_
n
<
T*.
The
emainde
o
he
p oo
is
a
echnical
e inemen
o
ha
o
Chacon
and
Hou.
We
shall
omi
i
he e,
as
i
does
no
in oduce
any
essen ial
inno a ion.
V]
Acknowledgmen s.
The
au ho s
wish
o
hank
Maca ena
G6mez
Ma mol
o
he
aluable
help
in
ob aining
he
g aphic
ou pu .
REFERENCES
[1]
C.
ANDERSON,
Obse a ions
on
o ici y
c ea ion
bounda y
condi ions,
in
Ma hema ical
Aspec s
o
Vo ex
Dynamics,
R.
Ca lisch
ed.,
Socie y
o
Indus ial
and
Applied
Ma hema ics,
Philadelphia,
PA,
1988,
pp.
144-159.
[2]
C.
ANDERSON
AND
C.
GREENGARD,
On
o ex
me hods,
SIAM
J.
Nume .
Anal.,
22
(1985),
pp.
413-439.
[3]
J.
T.
BEALE
AND
A.
MAJDA,
Vo ex
me hods
I:
Con e gence
in
h ee
dimensions,
Ma h.
Comp.,
32
(1982),
pp.
1-27.
[4]
Vo ex
me hods
II:
High
o de
accu acy
in
wo
and
h ee
dimensions,
Ma h.
Comp.,
32
(1982),
pp.
29-52.
[5]
High
o de
accu a e
o ex
me hods
wi h
explici
o ici y
ke nels,
J.
Comp.
Phys.,
58
(1985),
pp.
188-208.
[6]
M.
BERNADOU
e
al.,
MODULEF
A
Modula
Lib a y
o
Fini e
Elemen s.
INRIA,
Rocquencou ,
F ance,
1986.
[7]
R.
BOWYER,
Compu ing
Di ichle
essella ions,
Compu .
J.,
24
(1981)
pp.
162-166.
[8]
T.
E
BUTTI(E,
Fas
o ex
me hods
in
h ee
dimensions,
in
Vo ex
Dynamics
and
Vo ex
Me hods,
Lec u es
in
Appl.
Ma h.,
Vol.
28,
K.
E.
Gus a sson
and
J.
A.
Se hian,
eds.,
Ame ican
Ma hema ical
Socie y,
P o idence,
RI,
1991,
pp.
51-66.
[9]
A
as
adap i e
me hod
o
pa ches
o
cons an
o ici y
in
wo
dimensions,
J.
Compu .
Phys.,
89
(1990)
p.
161.