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ALFRED
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
P.
Convergence Analysis of Block Implicit
One-Step Methods for Solving
Differential/Algebraic
Equations
Iris
October 1991
Marie Mack, Ph. D.
WP
3349-91 -MSA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Convergence Analysis of Block Implicit
One-Step Methods for Solving
Differential/Algebraic
Equations
Iris
Marie Mack, Ph. D.
WP
October 1991
Forthcoming
in
SIAM
J.
3349-91 -MSA
of Numerical Analysis
Oowey
M.I.T.
DEC
LIBRARIES
2 1991
RECEIVtO
Convergence Analysis of Block Implicit One-Step Methods for
Solving Differential/Algebraic Equations
Abstract We
:
will prove that numerical approximants, generated by fixed-
stepsize, fixed-formula (FSFF) block implicit one-step methods, applied to
smooth, nonlinear, decoupled systems of differential/algebraic equations
converge to the true analytic solution.
Key Words: fixed-stepsize. fixed-formula
block implicit one-step methods,
Bansch space, Frechet derivative, Kronecker product notation.
Introduction:
In
this paper, vve will consider
some theoretical properties
of fixed-
stepsize, fixed-formula block-implicit one step methods that underly our
numerical implementation
discussion
is
in [5].
Of central importance in this theoretical
the question of convergence.
We
will determine
whether the
approxirnants generated by our numerical algorithm approach the
corresponding analytical solution as the stepsize h approaches zero
so,
what order
The
in h this
convergence follows.
first section of our paper will contain a derivation of our
miethod and
- and, if
some theoretical tools necessary
FSFF
to prove the convergence of
these miethods. The second section of our paper will contain some
theoretical tools necessary to prove the convergence of FSFF block methods
In
section three
we
will prove that our FSFF block implicit
methods are
2
section four
consistent.
In
methods are
stable.
we
wil] prove that our FSFF block inriplicit
Finally, in section five
we
consistency and stability imply convergence.
will use the fact that
1.
Derivation of FSFF One-Step Block Innplicit Methods
Suppose we consider an
initial value problen^i of the fornn
subject to the initial conditions
=iJD
ii(>^o)
such that
X e (Xo.XfJ,
tI^O,>'fir>WR",
and
f:lXo,Xf,rJ
X
R^-^R"
is {C^, Lip) on the region [Xo^XfJ X
We
R
.
can think of the main idea behind fixed-stepsize, fixed-formula
(FSFF) block-implicit one-step methods for solving (1.1), as going
fromx^
to Xk+r by approximating the integral in
^k+r
v(Xk+r) - v(Xk) -
1
l(x,i^(x))
dx =
(1.2)
.
•Vi
are particularly interested in the class of FSFF block implicit one-
We
step methods analyzed by
{X,
I
Xj .= Xo + jh,
H. A.
rh e (0,
and generates the sequence
Watts
Xfin
iij],
- xo).
in [7].
j
= 0, 1, 2,
such that
analytical solution to (1.1) at the point
Watts considers the grid points
ij
Xj).
...
(1.3)
],
approximates
^(Xj) (the
Watts defines
a
FSFF block
.
4
implicit one-step method as a procedure which,
equation of the form (11), yields
r additional
when applied
values \^^\,
to a single
yi<;+2,
-,
yk+r
simultaneously at each stage of the application of the method where
m
for
= 0,
1
,
2,
k
=mr,
...
A FSFF block
imiplicit
one-step formula applied to a single explicit
ordinary differential equation of the form (1.1) takes on the following form
in 17):
2
a,jy^,^, =
e,y^ * hd,Zk *
hy
b,jZk+>
(
such that
and
1
<
i,j
<
r.
Suppose for notational convenience we
-4
:= la,j],
B
:= [6ij],
let
i
.4)
I-
:=i *'i
_
-,-
H
n: =
1,2,
.
Therefore, equsticns (1.4} can be CDnclsely vvritten 2S
di
1
1
-1
Assuming thit *^
Sir,,
=
e;;i£ts, Vve will
further simplify system (1.5; tc oDtam
hEZrn * iyk * 'll^t-
(
1
r
such that.
G
:=
a'
i,
and
^-A
-1
r:=[l.
1]T.
1
Utilizing the Kronecker product notation, FSFF block irriplicit one-step
methods may be applied
to a nonlinear, decoupled
differential/alaebraic equations of the form
Fd(x,v,AA),ii^(x),Zd(x))
B><.^^).Lj''"))
suDiect to the initial conditions
such that
fi
system
of
I
r [vn VV..1
:_i" 0»"lll|J
Fd:lXo,Xt-in]
pn -^Pn
y
p.
....
Pn
•»
X RnO
X
R"* X
[XO,Xfin) X Rn<« X R"**
L:
iM
n
y
•
:=
:=
Rn<» ->Rn<«
^Rn«
Iyd^(x),i^T(x)]T^
nd + na,
and the Jacotnan matrix [^F^/azj] is of maximal rank on (xaXfJ.
If
£ is twice
continuously dif ferentiable, then
generalized Picard-Lindelof theory
form
15],
it
can be proved, via
that the solutions to
systems
of the
(1.7) exist and are unique.
For notetional convenience, let us define the follovv^ing nr-dimensional
arrays \^,
^,
tjo,
and Zo, to be
&r. :=
Urn
Zo
:=
U'O = \Zd\
k =
mr,
:= \'ik*\'^ ,
and
m=
1.2,3.
—I ^
I
Zk+2^. -. ii'k+r^F,
Id\
-, Zd^F,
Let us utilize the Kronecker product notation to define the following
matrices:
P
:=
^r X
Q
:=
B X
R
:=
Dr X
In,
In,
In,
such that
0.
Or
8
ij„. - Pij~..i + hlri7:~
M
+ Psr~..i1
The second section
of our paper will contain
some
theoretical tools
necessary to prove the convergence of FSFF block nnethods.
we
will prove that our FSFF block implicit
section four
we
imply convergence.
we
In
section three
methods are consistent.
will prove that our FSFF block implicit
Finally, in section five
Q'l
methods are
In
stable.
will use the fact that consistency and stability
2.
Theoretical Tools
The notation presented
in the
previous section will
now be
utilized to
consider some theoretical properties of FSFF block methods. Of central
importance
question of convergence.
is the
We
will determine
whether
approxinriants generated by our numerical algorithm approach the
corresponding analytical solution as the stepsize h approaches zero - and,
if so,
what order
in h this
convergence follows.
More specifically, we will
prove convergence of FSFF block methods (1.4) when applied to smooth,
nonlinear, decoupled systems of differential/algebraic equations of the
formi (1.7) via obtaining the following classical result:
"Consistency
In [4] H. B.
+
Stability
^ Convergence."
Keller presents e general abstract study of methods for
approximating the solution of nonlinear problems
in a
Banach space setting
Hisbasicresultisasfollows: If 8 nonlinedr protiJem has a uniQue local
dnalytlcdl solution, and Its consistent approximating protlem has a stahlP
L Ipschliz
continuous linearization, then the approximating protlem has a
stat'le solution
which Is close to the analytical solution.
Keller considers the problem
G(v) = 0,
such that
6;Bi -> B2,
(2.1)
2
flnd
Fi.
(i
= 1^2) Qr8
Qppropriete Rfin^ch spscss. On 6 femily of B9n5ch spaces,
the family of approximating problems,
[b]^', ^2^'},
Gh(Vh) = 0,
(2.2)
where
GhiB,"^ -^
62''',
and
<
h
<
ho,
are also considered in
[4].
To relate problems
(2.
1
family of linear mappings
P/'B,
^
)
and (2.2), Keller requires that there exist a
{P^^, p2^}
where
B,h,
(2,3)
and
lirnllP^MI
(2.4)
zllvll,
h-»0
for all V e
B,
(i
= 1,2).
For notational convenience the following is defined
in [4]:
Iv], := P,hv,
such that
I
(2.5)
vV,
e B/'if v e
we
define consistency, stability, and convergence,
B^
(i
= 1,2).
\
\
Before
a sphere about an arbitrary point in a
Definition 2.1: The sphere
we
will define
Banach space B as follows:
of radius p, about u e B, is defined as
follOYY'S:
Sp(u)
:= {v:
V
6 B, Ilvul!
<
p, p >
0}.D
(2.5)
Now we
are prepared to state the definitions of stability, consistency,
and convergence, to be utilized in our analysis.
Definition 2.2
some
for
llv,
for
all
ho
- Whil
he
>
>
0, p
<
(0,ho],
and
I
1
with
GO
llTh(v)|l :=
for
all V
Suppose
on Sp(u)
e 5p(u), and
u is a
The
if
Let
6(u) = 0,
and
Gh(Wh) = 0,
0,
independent of
<
{6^0}
is
consistent
(4).
2.1:
h,
vv'e
of order
if
M(v)hP,
some bounded functional n(v)
<
and only
e Sp([ulh).D
(2.8)
>
independent of h.n
unique local analytical solution to system
The significance
Theorem
>
u e B, if
(2.7)
fanriily
- [G(v)lhll
llGh((ulh)ll
theorem from
w^.,
and only
substitute u for v in (2.6),
l!th(u)ll :=
stable for
Gh(Wh)IL
all v^,
[4]:
IIGhdvy
is
and some constant Mp
0,
t-VllGh(Vh) -
Definition 2.3
P
The family {6^0}
(4):
(2.1).
If
we
obtain the following;
ri(u)hP.D
of definitions 2.2 and 2.3 derives
(2.8')
from the following
if
fnr =nrriP w. e 9 (iuu o
and consistenl of order
and
>
p i
1
alt n
vvilh
GO
e (0 n-i
at
Let \bLi)\ dh ^teD'e
fn*- u
u.
Then
-
IIIuJH
f Of Mj,
i
Wf,ii
n^ndOhP,
D
•;
llfiliTirn ih&nrarr
of
(2 9)
"'
1
v'^
'.A.-ill
fn^msnii rlpfinp
COP'-'erO'=""!'"P
^^"'^ QT''''^'
*''''d
convergence as fcilows:
Definition 2.4
convsrgB
sale to
lim
i![u]h
-
A sequence
of nurriencal
appfoximants
VV|.,
e
is
5p([u]|.,)
to 5 unique local analytical solution u of (1.5) if
-"
(2.10£)
vv'fj!
h -^0
for p
0,
>
sequence
and for
is
p
if
e (OjIq] The
ail h
there e;;ists
order of convergence
some bounded functional
M(u)
:
of this
0,
independent
of h, such that
!|[u],
-wji imu)hP.n
(2.10&)
Let us recall the definition of a Frechet derivative of a bounded mapping
H(-).
such that H:Bi^B2.
Definition 2.5;
L(v):5i
^
^\-\
1 i
lv.'i:-*0
D2.
'——
if
there exists a bounded, linear operator
such that
—
—
ii,..ir
'I"'!'
~"'""
=
*-''
C^
1
'
'1
tzr
€ B,
ill v:
zi H al
appl-nU-g
FSFF
:.v.;"
methcdi. to
btCC.K
tht forrn
i.j£terni. cf
we
"itandir:!" prccec^re,
t'
UUwC U
loll
'vv
1
r.
'.i
w*
I
Q
1
'.
.
iw^|V
,
U
i.-n
;
.
Vf.
'..II,
......
AC
.
.
.
.
'Z' *.
Uwy
^.
V
L
1
1
C
'.
u
w> 1
1
4
I
1
1
1
C 'S
I
I
particular, Y.e
in
kC U
^'
I
C
k.>
I
C
I
f
I
^
w
<3
1
v
c :•
1
V
1
1
v. ill
C
Q K ^' ^
1
1
'
,i_r,..'h-'
'
'
.'z
Of...;. li-wvtf rrtvIlT'. uti
;i,ji.p!r-y-
•.•:
L.ii
:
w. lulLj uu...
some Kq
>
.
j^-
i
li-.
c;
ceo C ^
v >_
i
'.
1
l::e ^c.; ^c. -
i ;_|.,
'-
:
'..
v
|.|
.
ic
I.
0,
i':
^—
'-LI-
b Ui
C.
.•!!
i
:
!
1.
i
Mil ^ _ .(.c-UUi .-
wUiUiriwUvb
Ull
^fQV
vv^.
.
:.ii2
i.
.
w
i
,
\^\
—•
•
U:
J
^i-i-.t'fi.'ii
...L,,vn
•
*'ri"'
\
It
The
v.'t,
^
j
\\.
-
"n'''
t;_p, v'f^i
.[_
v \4
I
:cr
li,
-ir^^ m;
*
f
7) is n:i a
2.1:
Spheres: that
i::
1
.3i.i!:y
'.l!r
.
C k-
will proceed as in [4]
1 rr fn a a r, d t h e c r e rn f r o n'l
Lemma
l't
>.
consiiter.cu
vl 7),
Dy iirnple Taylor Scnes copansiotvi. Bccaus8 the staP'lity
dtter'Ti'ifiec
venf'Ccticn for general nonlinear proDlems of the form
I
5,. HI
-h
':
chcll see that fcr our differehce approiamitioni, oDtained via
vv^e
"li
Frechet derivaiive
Ihif. L(v) is the
=
i
-,rj
!
f ollCi''ii"-.''i^
Ul
cC-Zwit
;_],i.ric.i
.uS
iheoreiTi frc.T. t4, injures
i
C.
i
.
tii-r
I
1
'.{
iC^/.
.
.
.;
:.'__-
er^isttnce oi «
i..
..._-
tl.t.i'.j ci
•
1
1
v
Theorem
2.2:
zroir
cor:.'ste;-,t ::
_ = .....iu ^.1 .i^._
ThrH, Tor pq
,.
;
p
.Lii
.
vvitt-,
1
-
vvf.,
and
GO
at
u.
Lei Ihe nypolhciis. U) and (n) of
iJ.ri-
sufficiently small, and for each h s (C,
h^, :
In:,!,
if
pro^:l6n-i
Gh^v
r-.u:
=
The .nequclU^^
in (2 ;5) \i eqi..;v3leni to the foilovving.
Ccn/er'sncs
"Orui-" 3f
-
A 'jo, we have convercience
Ndw
let
'di
Grdsr of Conslstcncu."
if p =
p
2
1.
introduce the concepts of the local discretizcticn iVr^^iy:,^:
error, snc! the overall order of our FSFF block irnplicU one-step methcai..
Definition 2.6.
error
.'e
at the rntn step
T -
ff
T
t
T
v.-l^
define X,,
that
is,
t
"^^"^
1=1
'
1
'
i r
'''-
t'e
the local discretizdtion
(:.i£)
!
=
n"i
-TIT
,T
t
2,
1,
'9-
'
—
_
,
T
r
T
,.
*i
~ '*d
'
'
- T
^ T
n
isd
=3
'
lo c U'.i.^^-r' .L^.c! c.i.2.y .:.-:, w
dU
.iC
.
CI..
.li^-.
••
us.u L^i
i._i, i_'i,
lJ
.';l. v.,L!fi
1
..ri^
cTc;
iiw^.;G;
^C ^.^^
'.
.
_
vcIl.*;
1
_
I
.
prcjietfi
.
2i
;.!;=
;
i
.
;
;
lj
flJIB/floJ
locdJ truncation errors.
Definition 2.7.
Gr.c'~-:''.c>
IT,
i t h C J S \\'-3] IS
-
order
overall order
^r.r
-^'
'
-^I
'
3C^rd*r
^^^ ^-^•'
^-^' "Icck
uStWiid 8i
"^r-
•
.
.•
!.-•-. tl-.a*
l»
«^,
-K-^1
+ ri.'hK!*M
r
-J
..
w =
1.2.
,
It-.'
1
V
_
c
111-:
1
i
r V
C,
(
=
.:
I
*
-
1
1
rn p
1
fc.'
C
I
.
C
:
-.
t^
1
11
1
^j
.
f1
*^
*-• I
t
I
J
_
1
1
-J
-
II
I
I
-.•_
V
C
1
1
.- 1 1
Ci
-.
-\-^^-i,
U
1
-
.
I
;
;.:
u :•
,.
.
?
•'
t:;e cvi:--:]
1
i
:r :.?]:c:t
I
orj?r-X,,_ of ju- r^r
_-
.
-
.'
F3FF Index One Consistency Analysis for Smooth
3.
Systenris of Differential/Algebraic Equations
i
j:E, -^ 5^,
::= :::2i.
difinad ^n (2.1\ ccrrtspcr.di
U
-dV
i'l
1-1
Y
=
I
;
F
—
/-. 1-.
*
i-
- p
..
_
->
.
!v.
'.'
.1
•—3
:.'
<<•'
I
•
.i*^
l"
V
••••
I
~
Ir
'l
M TIT
F,,.J
[v.
fv.
•..
^
!
.
•..•.
._ r,,T
Ti-iC,
V
I
'
—
1
^i'""!.!.-' •Tin'
...
*
'
5
;-• u- Tin-
u
il'
I
P ^ ••
''--'-'-
•
pnj
—
pno
f^
^ TIT
s
;-•
»
C;
^
;
-«---i-
-1
;ts 0'
'-
-
•'.*.»
"Tilt*
t'-.e
"
norm
c
te define:: in (-.2;, snd tf£
--.-.--.
-
-
r-
--
V
.
fc-. 1'
definsd
V.
which
- l^c]
c-
uL
I
i
'.lie
1
':'r
,1
-143]
'
','r
-nd
I
^dj'*-'"'.!
on
••o.Mf,r,].
The
R'"'--
••
i-
•
;
I
.._.,.!
'
We
will denote this B^n^ch space
elen^ients of B, are of the
form
'
I
-ri'
•
• 'Tin'
\.. ,!v- v.. 1
-Ha ]•'•"•'-* ••••tin'
•^1
-.1-
'.
-
-H3J
'
i"''
R"^
f^^" -
P'([vn.^-tini
-.
Dnd V Dni v Dnd
c" c^'^ss
6''e
..........
^.^
and the set of functions
—
1
''- taj.ij... .^o*-
*'
*
{2.2).
'.r.
-tv.
'.'
'*
I'lri"*'
i»--ij
.•yip,
.'v.
v..
l^
J
1
'
Pr.j
"^
.•
-k pr^d
lu
_
!!
1
£
II ::''l
-^
=
:=
Bi
(i
=
1
suodlv!!
,2),
then
!!rii
we
will let the norm of v be defined
}
:=supltii4dlli„J!y^.ll!„.yi
sup{{ su?
1
iiind
iy.(K)!..
sup
rid+1
i
A in
!v^ (;)!..
-ud
lilind
|r,(:01}
:
x e [xo^k^.^,]
a;
n
._
_
.
is a
if i;
solution to (1.5), then (3.2a) reduces to
:|iJI:=sup{iML.!|v,n
u
'
«,,}
:=sup{l!v.I!^^J!v.||^J|v.|||^^}.
(3.;t)
Lbt us ccnsidsr the folloyving family of grid points:
:=
y.^.
Xq +kh,
iZ5}
such that
% = nr
and
£
m : n.
Because
of the
K
mgnner
in vvhich h is defined, v/e
see that
tin
For notational convenience let us define the following r-nd-dimensiunsl
arrays:
Uind ,rf, ]:=
[iiJ
,k+ 1
Uind,0V=lzj/.
"''.•••.
ild >:+r'''
V
-.id/]^-
2(nc.m] " iSid>;+1^. •, id>+r^r.
Zjndo]
ror
=
'=^,0''".
-.
ixi.o'^]''''
,
Also, let u: utilize the KronecKer" product notation to define the
foUovving
n-iatrice:.,
which will be utilized
correiDonding to the differentia] variaDles
denned
to extrect the equations
fron"!
the difference equation?
in M,9):
and
=
P[r,d]
Where
X
!^'r
l'\, B,
lr,d.
and [v are defined as
equation
in
For each such grid of the form (3.3),
we can
(1.8).
define the following difference
schenrie:
[nC[pH;]"'
iU-nd.^l'^NiyU-l -t''C[nd;2(nd,m]"'P[nd;Zind,m-1 V^
ija^'-kri..
io
,»:+) ;i<i,K+i' -sdl--*-!-
- iV-nd,
1
-
.1
I"!'
h^'m, Ti'Qp
-Tt-P,
,/ip
'.
p Trrir ji + Pr
j"'**!
M
-•--'•,
:•
.:
rr-,
\
1
1
i
<
r.
The Bansch spaces
defined
in (3.5),
B,^' (i
•Is'
• il
•- -
~ Ud,k+i'
iVaO^k+i^ id,k+i' liU.k-^r - Ua.k-ri'
C'
I
= 1,2) consist of the
norm
to be
and the (n-^nd)3<.-din'iensional Euclidean space
-'.•-tL
.
.
'
(R^'-
;;
R-^
The elemsnts cf
R"-:^,
>;
I
h^
arc ci
B,^
fen-;;
sucn tnat
-
','
...
,,T
['•.
_
.
r,,
'''.'.1
"
M-i.Oi
JIT
„',
_ '
MT ,3] - '''S.nd+I
..,
]T
,,
'"J.nd*
'
rr
..
,
•• '^s.ri'
•
>
V, € R^.
V, ^ R'-,
I
!:
!l
;..
|!
,i.
ij.,
e
P,*^
_ .r.siv^:
i"na,-,-i
!'"
y
.2),
iv<.
in;
. .
lij^ni
.
.
then the norm cf r^ will be -efined a;
)i...
"
u<if]
.
i
,
I
v,"
i .< iri
.
i
'
II
i
ii'ia.', i.|_v^i!
rr;a:i:-!nq
6.?,'
correspond: lo the
__
v_.-
<
(
,
ii^.
.
—
(n
.j
z-j'
•••
tI'
..
I
•:i<i!l]f\i?.Q
nd''X-
ivr
lii
follov^-s;
V.l
:
1
i
s
^ 1^
.
J.3
i-;d
:,1
.
'
.=
.:
1
m5.^
\.
then 1% =
1)
1.2
II..
.11
max
,i,
•
is a sOiu'-icip 10
t
Tne
11
1
1,1
(Tia.-;
'
=
(1
i.
r
t'of
» .
1
i s
i !i<,
a no
<.:
5 a.' reciute':
-T S
1
\.'-j:
3 1 A.I-
in (2 1),
ii
and applied to
d'screte e^u^lioris defined in (3.4).
;
:
C '^ L
I
.r
1
\. 1
>
Gi iU Gw'
.
.
nj
'.
'iV
I
f
I
I
III
ihu^i^
iCu:
i
|i^K
1.
I
.
I
'
L'C Oi iw
I
.
f
i;
M
_
-
11
.
li__
iV
1
1
,^c r
Alio, bccsuic
gria u5fTr,£u
:.y
U;
..
|
-
I
,—
.'
.
t Jl
i
Sep ui li_
vu
-1
Ic riiu^,;
;
l;^
i
'.:
.•
ci-
the
,
ir
II
J.:_.l
V.
.Lj;
n
--.-1,,*
!f
,
Li^
._!., l.llCllV.-_v_^,
Vv'c ii'c Vv'crl..]n;4 \iitl'i
(3 3)
C.i>J
V
-.U-. .CUUwb^ lb
srnooth func-t^ons, vvc
Decomej denser, the meximum value
spprocch the supi^emuiT;
in (3.2)
Thus the limit
f,i";OW
inat
in (3 4) vvi]]
in (3 7) ejnsts
3!
Proof of Consistency
are equipped to prove that cur FSFF tlock implicit one-step
Nov.' vye
rriethodi (1 9). applied to stricoth, nonlinesr incex cne iyite.'Tis of the fcrrr,
(1.7), ire CQr.s^£t£nt
Gefinition 2
with
GO
£t the soluticn j:
:=
[^.z^'^V - ss defined in
Using notation similar to that utilized
3.
in ,2.6
Yve Ojtcih
),
the fc'lovving
=
.'
'.«
'.11
^
<
I
W
[^^
-)T. r..{
r„(-)T]T^
.--iT
f
-.
1
[hC [nal^
I
i
.
,rn]-F [T,iUr,T)
I
^
'ic C2.16;.
T.
li[r„3
- [-
whe-e
,
,T
T
,
TIT
-h! D [r,v2fnd
.rp]+R [rri'Zbd .m-l
]!
I
T
,":
1
.
1-.
—
Mil
1
<
1
.
S'-,l-!
<r
Taking
r,r:r-rr-,s
of ticth sides of (3 S) yieldi-
such that
r.
•,.,
Qirid.rr]
"
^i*j !.••
Uirii.rfij
-
ii4.3,i-t-l
Vv't ca;'. riOv,'
T Li
1
1
..
r,
'
••
Ui>-i-r
sj:7"irr;anze
our .^SFF
ir.de;;
one consit-ttncy resjlts as
z-
Theorem
wuiic-iuti
3.1
n.ni iicaf
t-^uations of the tirn'i
.jcivi-ui-ii
.
(
SiTiOOthr.rSi- DTD'-BrtltS
- c
TIT
tu
sy^'-ciiic- oi
i
ti
oil
i
a.* c.hCi.-^ g.-.
17), which satisfy the aly6lrc'iC and icca.
dliCUSS^d
L*t
unique ^cia! analytical solution to (1.7)
Suppose
ui
rFj/.rVj :s of full rank on
[>:,:i;;f,n]
-
Ws
ui of classT'-*'
Vvlll
or. ::.o.:'fir.'
arrange to solve the
noru^near roo*
constants
c^
4
-'i
'in',.
-i"
I'n-
I
>.'!
>
.
!
I
I
-
and
.'
1
1
*
1
U c
I
,'
..Ml
ii.it^ ihz'.
there
sush that
t-''*i
^ f. n''*^
- ^-a
»
,
i.
Then there
jU
r^,
r
'ill. ,3, M.J
prcDiem so accui'ate'y,
finc;ir,9
/
e;ri£t£ an
M
l-,r
M
>
0,
indepenaent of
h,
such that
e;.*;:::
pDZKive
1
FSFF Index One Stability Analysis for Smooth
4.
Systems
Ncvv vve will
methodi
of Differential/Algebraic Equations
prove sta&ility of cur FSFF block: implicit, one-step
(1.5) cpplien to nonlinear,
Gecouplec
c;i:ferential/sly6l:ra;c equations of the
form
c;n be summarized as follovvs: The family
in
equations (3 4), will te ccnsidered.
Frechet derivative of
S^,
Cur stajilULj analysis
(3.1).
c:
It v/ill
one systems of
indej;
mappings
be shov^n that
where x
at L'l,, exists on SpoCUi,?,
satisfies (2.12) and (2 13) - the hLioothesis of Lemima
To verfy
(2.12;, it will be
shewn that
if
Gr,;5,'"'
'll}n
^
b2-\.
IyMiW^
:=
constant M
:i!//;^.!i
^
end
C.
-=•=
[jJ,LyV
^: £
2.'!.
i^ the ".niqus
solution to
UCl^lhMATinzij^'n.
(v'her't \il]fi
define:
(4.;j
'^ij^, will
oe defined shortly), arc
if the'"e
ei^ists a
such that
(4.2;
inlllj'i^/!,
then
that
IS, L^C^lr,
for- all
h€
'-'
IS
uniformly bounded at the center of the sphere S^^'UV'
(:,*!.
The secono nuoo thesis of
satisfied
if it is
Lemma
2.1 - (2.13) - will
assumed that the function
second partial derivatives
Vv-ith
F
be snown to De
has continuous, and bounoeo
respect to v and
z^.
With
all this in
mind.
.
provt itGu-lUy
v/t
in
Gil
it:"en*iai/Giuctr5iC eqi.ciions of Ine
ncv. tq;;ip:.e^ Ic
o:
tl'ie
forr;",
FSFF
(3 1)
oloci: itniiiicn
or,r
Derivation of the Frechet Derivative
4.1
Cir-r-:^ '•?
(II
/
''.'''"
^-
-
-
"K
*h'if
"I
lit''
.^-
'
-•-•-^-'a^-
I
s,-..::
Tnc Victor
Silisfis: thr :cillcv;ing difference scheme:
ii'h
'''•^[nd]'"' 'llqnd.mr^'-^inijiy. ipd,nn]"^'[rdjli:!fr-1 "-"'^'(ndllii.
'...-
r
^
_
''1 /'^
.-
Y/here
•..
-
L«,
-
-
l54nd,mj
tr.
T
r,
,
-
'.^.k+i
'114-1-1
_
r..
I
^
k =
ana
I
-
- lUv,'>-r1
I
t
— ^
rrif.
fc'
,
.
'
•
r
'
•
T]T
sJ.k+r
iiv>.Tr
'
TIT.
ILft-r
..
••
~"T
jjjj.fr+r
...
••
'
T'T
',.
,
T
''"
~ '—d,K+l
.
T
T
r
Mna.rti)
a,..',m
„•
.1
.,•
'•...'
-_
,
,
is- ir.o.mj
(nd.m-1)-*IIind.m)=ir.nd'
<
TIT
''
'
'
,
1
.
Ttirz
r
-
1
li c rcci Ju3l
term,
Cifference scheirie (3 4)
rrn^.m)'
^l^nd.m] "
F rv
;
- r
1
^^-^ because jih does not satisfy the
e;;actlii
S«t: raiting sjsten'i i4 4)
if'^'inci'"
'f'
fro;t^ systetTi (3 4)
'''^indjii ind.m]
.'•
.•„
^
_
yields the follovvinq;
" Pindjilm-I " ^'"indjli (nd,m-1
jJ
=Lirid/rtl..
'-,
U.5)
such that
!'i-H'-Mj.mj
- 1'- t.>1
- iiirio.m]
aqrid.m].
ii[rij,rn]
~ Idiraju] ~ !iL[uij\i],
li
= tind ,m] ~
Irid/fij
ilrr
liirn^.
--
/•n
•-
—lUv KtI =
1.
]
h".
n
.
-
li'hd.,nr
+
ind ,m]
F,
>
'At
iir.
iJ:^
i
~ Uv; >+i.
p.
r
The Frechet derivative, Udx'lhl of the mapping
lir.srn-^--.,- f^^'i
thg+
ic.
G^,([rlr,) is
obtained tq
r\j
,-
1
•
-
•-,
''-.it
«
._,'-
k =
'Li
I
rr.r,
W
11
1
..C-.
r.
'.iMiii-f
1..;-
•.litr-
1
ii^.i'.
(luUw oluC
!-
1
V'T-.'/
(.0
ii.Ot-.C
(.tit
ui
l-j
'.
.
1
1.
.j
ucfiniriOfi.
Definition 4.1
C
r
.
'
•
^
1
i
.•f..-.l
I
r -.
c
^' ^ L1
1
1
I
*
'•
(. I
-
* \r,c.
LsiUitJ
,- -.
-^
-"HQ-'i-'fT
-A
_.-•,=.,,(/
•,
The Frechet derivstive.
UCIiI,,), cf G^,(Iry
cr^
*:
u-5
,'I..i
/'^^'•'•l
,
l4~!
|1
,i
I
'w
'-
u-r-'
'*•
f
•!
1
i^'-JJ
!
I
vM.r„;
It
Fi^r' ,.!
r_i-,-.
;rA,
?l,.. =
ro
,1
--1
[.•«-
t*
..i-,J
"^
L'
]r^[r4\^^''[r4
-J
-
rr:
Urnrri'j
k+i
•J
J'+i
ad
=
k+1
k+-.
J ad
1^+i •Or,a,r..i
for
m
<
1
i n,.
k = mr,
and
1
1
i
. r.
Also, rec!!'' thrt
for
I
i
We
m
.
E'''
is
H, and //n
=
the bsclv'-'crd ihifl operator that
n
obEer'.-p thai if v/p ij?''inp
is,
arid
then the egi-sticns w\ (4 6) csn be concise'Li rev/ntten as
SLji-terr;
C4
1).
Proof of Fundamental Inequality
4.2.
For notG-.ioric; corivenience, vve
(4.2)
Fundamente} InsqusJity
our
ge.
vvill
refer to the inequality detinec in
(F.I
true, Vve Y/ill also have satisfied (2.12) of
)
By proving that our
lemma
2.1.
F
!
Utlizmg (46),
li
v/e ;•
that si^Ltem (4.6) can be revvntten in bloc^: matrix notation as fOilOvvs
--
^-TTiii-m - %r\'
iMVii that
!fX[!r,^n,d,:.ij
I
Rrn
m
=
%u
L,.;,
/^r,-,.i
+ i'm.h
P :r,^| li--r^": [r..:! U:[r,c),m-1 ]*•-:•-- Onu/r,;
"Dr.
P[r::]
!
hP[-::
-rr.
Tor
1
i
rr"!
i
?^
n
',
.
utnize Elccr; Gaussian E;Mriina::on to solve susterc. ;4
u:v,- \:i vviil
iljnd-'na.m]
Yi'e Vvi'.
-
-
tL('-id+ri5,rrf]
rr'
*
'•""•
*
w' '^
•--
1
'. f
I
•
J
-
ii[r,3,rri]
n"iatnx ecjotion
f cllO'vviriQ
7"
1
m-
'
1
'
ilind-^n
i-^in-i-l .dd]''
j
.m- ^
]
'^Q^.hd.m]
'
-Q^.lnd.m-l
l'""^'
-!j£_.ln3,rf.l'
i
'.4.10;
'
•-
lrr..'~f'•I''r^^^l*•.
-hflr
•"':'
i/yr..
..1
Ir,-,
/fl
/y(T,,ijl
/>.^1i'^.^%..^•/ftr,-\
/ n-;-l
>^.
im.da]
//•
-
im
~
—
'
s'-aa
'""da
I
'
*'
l+r"
-.3d
"
• ~a3
''
J,
.
,
'
>*i
,dd
M,
Orna
rT;d
"?"Jdd
- "•=-'
'' iro.QGj =
-
M
..
~
im,.33)
'
hP[r..i]//:---1,d.5]
_ _'-i-'5-fr» i-t-n-iri i+n
'
*'
r".*dd
^-od
.
'' irn.ccj
"'
•-'I
;=
Crni
,
=
P'-.r.dx^-nc?
^^
R'-.nd.-T.'-'j
f-r
''.•
•^'ind]C(nd,m]"*' '^'ind)"^[m,dd]l''
t'nr.i]
t
ili'nd.m)
sCiVe the
1
"'"II
'
>
;;
'
•
'
ri
k+ri-iri t+rii
'
''•
^"60
'.»dd
'Jod
'
^
ds
"•
yV'
•
•
"
•
-'
M
e
P''.ri.axr-.rMJ
.i'JJ
•
•
[n-i.as]
"^
'
II
Pf.ndxr.nd
j^' e
?Vr
^ "^
* [m.ddj
>
P-
-
.'
X
-Ilv:_.[r,:,r,-)i
5ind
1
m
i
<
I
= i-^%_,d> + l''''
n.
•
>
^Il^.dMr'''^^'
V/e vvili then utilize the solution to (4.:0) to obtain ir[ri^„,j.
that IS.
for
1
'AW)
=^[rn,dd]li{nd,rMl * >^Im,dajii{r,3,mri-'^[m,ddir' •^Q:^(nd,m].
li'lnd,--;]
. rn i
n.
Becauie ^J'^^
\i
nonsingular, system (4.10) can be solved fcrli^'r,^^] as
folios
such thst
f
or
1
:
i
i r, k =
nr
and
1
i rn i :M.
Subsystem (4
1
2) is then sutstitule: in^
(4 10) to oDtain the following:
= ^^[Pdl * ^^(ndl ''^im-Kddl
+ hllf^,(h),
"^[m-1
..dsl^-^Im-l
..aa)!''
-^[rn-l .adl- ii(nd.m-1
]
(4.13)
)
that
;.ucti
=
rL-,'..u.-
,C'[r,^]r{^H,rr.]
^ "indj "•
i
«
^1
lit
"— Iir_.[nd
n-'* im.cd]'
w(r,ijj
'
J
— Il-_.[n3.m]J
"[m.ds) ^"[m.sa]'
— Il__,[no,m-l ]*^[m-1
im-1 ,Qo]'
^
"*'
rn]
'^tm-1
.da]
— ^;^:_,ln^,'n-1
.aa]-
I'
hui vve ottain the follGVv'.ng recurrence relationslv.p:
i'l
i
iuch thct
.^
J
._
......
^
in''"
ff,.-
'
-
*
W-.
•'
-'r
'•,-,
.
*c. iOVi
•
lb;
I
.,r
"iTii^
m
_
The
.;.
'•
'''imoQj
''
"IrD.ao;'-
''' Im.aaJ*
~ '' (m-1
''indj ••'' im-l ,dd]
'-;nd>
t^. .oa)
.oa) '''
" iTi.da) i"lrri,aa]'
'"trn.QOj
'' [m-l ,adl'>'
(m-1 ,aai'
"im.adj--
Drri''-'
'vr
,'
1-.+
1 ,- ,-
•
r,d
|i'»'^
_.i._,w.
.
-incj
incj
!'1J,
I
••
...r-^rj
.
M
''f-
_:
+ i'V
,'i
I'll,
;:j._.:u.iai wUi
;
v
c ^3.-..3r-- -V
H.-«.>--.
:.
..,,
,
i
f
i
'(•"I'l
i^^fc
,,v »•.-•,-•-
,
Vi
'.A
s
'^'
..'
I
i
i
i
i
C
.
...l-'i^l-,
li
,i
i ^^i !U
i
u.
.,
'^':
,„:,::i.
c.i*i-»'-- *••-
cifT&rer.ce eQJ5*.ior;;
.••
Tp...i..
- r._,
.1
C:.-
nrl
.
.
f,
i
.m. .u.
.ijT> •"•' "
v.
fcll.::.vi.'.q '.ernrng li.
necesionj to er.atle
ui.
to utilize the
sequence
!
'
Lemma
4.]
11 'dit ir,;lial
"1
•
,11
.
'iiiriO.rri!''
•?•-,--
r--
1
>'
-
Proof of
The
for
£G;"i",e
^
'''',
•
Lemma
.
L
"
"^'
vvill
.11
"1'
•'iL(nC,mJ''
••<•
'
=
'
-,
uO'.v
= 0.
TT |i^
''
i''
•
ii=1
+
"V
M:_Ln.|-l
>
I'
we
i,2S_i W..''ii
ii"
'
i tTi i yi.
is triis
knov/ that (4 16) can
',
1''
r'^ "*^
'
'•
i
P-
for
.
I'f-V
»H'lt
-'
4_ i.=— -
s=1
h
'i
TT ''M-ii h
hiilir*"
I'
I
!
1
II-
11=1
,'
I
i
m
-
,
Eeiauic
suppose (4.17)
as fcllov/s:
V^r,
J-i''
Term
cf firr.te difference equationz.,
t
•rfi
~'\
i
(4.15) and (4 15) impiy
('(•'I'
From the theory
be proven by indu-ition ai foriCvVi.
(-^l^J i£ *"^£
Thei'i
-'<>• Tn-r\
J''
be scivei fcr
'
4.1
(4.17)
0)
?l
."
'iiinC.m-rl
1
^t
n
ll^r,^
m
;;.,
C
'''m-
inf.qijailtLi
Vr,- 0, a^o
for (4 14) are coGClly satufieu - thsi
.'
'.
-
cG:".dit.;ur,i.
.1
-^
Ii^L:-'''"'"
11''
'
i"
*
••
* Mli".vi.J,r
m
I
^
-
I
1
1
1
un
,
'
vnU prove
Nc'vv vve
Lemma
J
^
I
I
n
1
tf-:2t
II''^^''
'
' ii
and
!;r;-^o^Tr,:',
l^.^,
I ^j,
tounded
'or
I
4.2
l^.^,
l^^.
and
L^.,
such
tlial
- '-i'i'
'I
lll-'ij
2-e
-
.>
Suppose there exist positive constants
'i-'df
i;;K^,';h;!l
ll
J
-
i 43,
and
for
1
;.
]
hil.
r, ar-ij
i
r
=
mr. Finatly, suD&ose that
A
is
smaii enouoh, sucn thit
),
Yvhere
h 5 (O.^j,
and
Then there exist positive constants
\%i.>:
and
fi,*.[,,
such that
Oi
lij
IIV
''•,'''1
»-|
.
we
Frc- iTi
(4 14)
H
l!^ v''
Lemma
Proof of
Firs*
"
(d?''
4.2
vnll tackle the inequality in (4.20): that is, let us oDtain 1%^^.
we see
that
where
'•2
for
=
1
i
Lf
.
''^[nd] *
m
iw.
:
.
•'
-'[nd]
'^(rrrl.odj "
^[m-l .daji^im-l
,ac.)^''
^[rn-I
n
.^;-.T
2
IL'ij.'.
q.
i.V|
-..i-
fecund 'or a, vie relating the
i.,i..
10
norms
'^'v;.
i^j
,
W-— ^y-
f '.i
of the varioui
Vvt
r.
matrices
i
i
-
I'-
*
'nd'i
(
-
L«u
^d
-
''^'''
rn«f'''f'l
t+ii-IMIH
Mill'
I
1
/
i
.
r
'
-da
the hypothesis,
''''im.ds]''
- "'-'•'^?''Jdd
'^
'^da
'
'^
•
•
'Jdd'
'
'-da
•'•'
-
.
in
LipschTtz constants in our hiipothesis.
''-mo]"
,ad)'i''
rv'
i:.',Y
cv,
'.^".-21;;
'_
and to the
^3d
ms'^'-'
<
-
iy
:;-,£
I
'J ^-
!«'JVv
ViC
[;2a-
ji,
.=
o
-
-
1
i.
-d:,.
'+11-1
'
1'-
<
•
1
-
r
I
I
•- >.'
!:!
h^R:
I
for
-*
jT:
*
;^:;i/(i -
1
-^
r^.'-i
^ad'
iilr
...
-
\
m
-.I.t
. ;:,
. 'Y^
!..;
.W.Gil
;.
|.'^
,
iiiliLfi YV
1
1
1
ij3
L..iiitcu in
•--.;
.».-..-
hJl
h?:-.
Ulilizinq Ih; individual bCLinds for the
<
_
vancus rorms
in
c^,
,
(4.2T\
a^'id
hn-;!,)1l
i sA|ji.i v
I
+,j^
;
x.i,
',-»--••
.
for
m
i
1
V
n. The following norms
.
p.
_
,"
/t
ii
Y
I
II
_
"
I'-r''
11,.)
-
II
vvill
be utilized to complete (4 25)
1
and
*
ii'^jnQj'' •" ''-r
i;4.25)
Thrrefore
+
i
...,
f
f
1
u:
.:
t
..
I
n
!
-,/7
I
!
!
.
•-
+
I'l
,_ •|U IIjII ,n
.
+1
.'I
1!-"'
c,..r,'f-
.•
~
I'ii''
can be completed as follov/s
II -III
i:
TiO'
TV
/
I
/I
'
^
V
/Mpllt
/
nr-'i
!^'.>h
f
J.
Therefore (4 22) can te completed
,,,
lOr
1
<
i»^i,ilJn
m
i
t.-.pinv
i
*|j^
,
a;- I'-'i-v.W W > C*
i
I
I
i'
."Ci-b-.p ijldu.,, Jt/ itDi.j
K Thijs (4 20) can Ic completed as follow;
i
ri
«
1.
'
I'
QJIHiUi''!:
^
expi(m-i)hR
1
evrin[(Kf,,-^:c^^^l'R'1*Pi H!^!/!:B!!}}
{]+(^i+iJui|/'i!Dll)}
1=1
-•
for
!
:
i
i
m, and
M
1
k--.-;
i
m
<
>1
Lomms
Proof of
''4
'ror.'";
Vm
(-fir.
=
II'
J
'n'!
-iri'T
rs-imr-lo+cc tho
Suppose
/^[rr.,33] li
i
1
II'
/ inllfW*
*i^rWlKj W'
yi
(A2r)
4.4
i
N,,,b
hsvs
f^tl!..
Lemma
1
'W.^f'
vv'5
n.D
i
Ti'-i fnlln--^
for
42
- ^:o)f^'5i.b''f'a-(l .n.+.b) li'^
C-iK^
Hm
f or
!9) 2nd Ismiris
^ !^-.t.b
<
4.3
r
and k
=
nrnnf of nuf p
ncnsingular for
1
1
m
\
i >t:
M 2) fCT iTidSX OTlB
that
mr. Suppose the hypothesis of
is, J^^-*^ vi
lemmas
4.
1
,
3'JStSITlS.
ncnsingular
4.2,
and 4 3
are true.
Then cur
—
F.
-.11.-
Is true for
•
aH
Proof of
i
!.
*
1"
.1
(O.Al.D
h e
Lemma
max{|i//„J!
1
.
m
4.4
1
K}
:=maviiiiii^T^ir,^^^^T]i|;
£
mav(
mav{i:ii^,,_^j!L
1
.mini
!ltlj,,^„]II, llti'(nd,m]!lJ:
^
^ ^n
<
n}.
(4.30)
Now
let.
!l>i^(fi)ll
us otjtdin hixati
he-
^''-''''i
''^
14)
heve
vv'e
!iti,,(h)li
!i^(m,d.]l! !!l/5^[m,.3]]-Ml !lAll^(n,,rr,]i! ]*IIP(nd]!!
1
!!l-^m-l ^ddll"'
I!
liQ^N.m-l
]!i
*!!/5(rn-1.da]HI!l/5(m-1,..]]-MIII^TJ^[n.,m-1]II]}
for
1
<
m
1 >:,.
Now we
Ihui proving (421).n
vv'1]1
obtain a bound for the expression defined in (4.19): that
Lennma 4.3
If t;'.8
constant
for!
:
hypothesis of
C,
such that
m iHU
len'in'ia
4.2 is true, then there e;iists a positive
is,
Recall
trcrr; (4.17),
and (4 23) that
(4 31)
<Cli(i'V<!!
for
1
1
m
:n. Also,
recall
from (4.12) that
^ !l[^(m,aa]]-^ll illi!k.[n3,m]l! * ll^[m,ad]ll lllknd..m]lii
for
£ rn i
1
H. Finally, recall from
lili"[nd,rr,]H
-
li>y[m.ddlikr,d,rn) '^[m,dalii[na,ml *
li/Vi^^,,,!! ilUr,,
<
= L,,{1 *
f or
1
i
m
1
(4.1 1) that
,,3i!
^ !l/y[m,dall!
L,,r. L,,Z
!!liina,m]!l
l-^'^
Im,dd]l'' ^H^.Ind.mji!
* IH-^ [m^dd]!"^
,3(1 -L,,C}!|!il'
!!
!l^Il^[nd,m]ii
yi,
(4 33)
n.
Utilizing (4.31), (4.32), and (4.33), vve can complete (4.30) as fclicvvs:
IKAfW! <max{C,i,,{l *L,jC},l^d'i
.=
MlKr)nilG
'W-'U,^Jy^
^U^CljjIKr)^.!!
(434)
1
4.3
Proof of The First Hypothesis of
Lemms
-
2.1
Uniform Boundedness of the Inverse of the
Frechet Derivative
Now
that vVb hsve compleled the proof of our Ft. (42),
verify (43) - the first hypotriesis of lemrna 2.1
Lemma
r..
..
t
i.-.ri ,..«,., 1.
eqU J . ibl.c.
Ui
".--,-•-- he
0;-^^'-c=
as follovv;..
nornir.ear, decojp'.ed susterns of differential/
.ilC
;
Ui
I..
.-
.
I
..-,
1
Ui
VV III
^
1 c 'Vi r"-- =1
i
Oi .biiliiiai
;!m-«,;.diii
t-,
:.ilc
.
.
.-
*!•
.- '
.-.
-•
zl
•4.
wli
•''•'*
1
1
li.I_3-'
,
C^ji
''
7
"n.*., '<-'.,
IC'
3>-i-'
Cilu
Then the Frechet derivative, ir,Hi:\X ^^ ^/^^V- satisfies
hypothesis
can proceed to
4.5
SupDCse we consider
l^t^. dlL
-
we
ct lernrna 2.'. ihai is, the mecivjaiity
Proof of
Lemma
The ineq.,a:ity (4
{A.-i'i
4.5
3;
follows
fron-. Ocir F.l. (4.2).
is
-'I
I
-.
.
I
' < «i>"i
'^.'^ C:
c
u.!:, L:.
*r'ii
li Jc.
(2.12), the 'irst
truc-u
Proof of the Second Hypothesis of
4.4.
Lemma
2.1 -
LIpschJtz Continuity of the Frechet Derivative
To complete our
4.6
buppose the
iiu _^j U.I
.
is
}.
]
Lipschitz ccntiriiiou*
LUC os-UIiiJ ^ui Liu;
widi-ijwC-s
is true,
_
"
.
{i-
u^^i
ival;vCi
c-i
vvit.."!
^;.r
that
,M
.'
I
V'
'
there e>;ists a constant Kl
is,
!'"
I
-
>
0,
•;
k,
-
-rO
'
'.i.
'n
••
'
and
Lemma
Proof of
4.6
The vectorJlih satisfies the fclicwing deference
•"i-.
.'-' '«»
r..
i.;i c.^ri.2;'
J'
,iri<i .Tf:
r
i-i.iriO.rrij
•
r '•
J_ij'-
where
.
.
••r+l
.f|-
._i-..-.
Sl.)
;^:._
r_
_
-
1
--p-
.
'-iri-3jlt.j,iri0,ri"ij
'
i'-
ii" n.:.
.••
Jj-l.k-'l..
...
\
_ r
2i_Vf+''' - —ill
J
.
;
..-..•>"•
-•
.\.t
iriOjUL.! ,rri-1
such that
I'j
II
T.j;
I'!
reSpeCt tc £
tc iiari:! z^ are ccntinucLiS.
Ther (2.13;
.'
in
TuriCi.iCi". r_
,..|-^pi.
i.-.(^,
With respe:t
VI
second hypoiheiii
that is,
of eiMrr.G 2.1
Lemma
ils^.tiity ar,cly5is, vve vviil ver;fy the
-i^D-
schetTie
-LI' r
'-"indllk. i.lrio.rn-l
•'
j'
"='•
,
|.^,.
;;:
,,
.
A
iLI.J
UN
Wf
.
kt
_j
=
=
T
,
..
T
liU,.3> + r..
..
'i"'',
...
T IT
,
_VVj^K+t'
J'
,
TIT
...
..
,
[rj,d>:+l''',
,
lij.d.k+r'j',
,
iv.i,kfr
F,
,rj^a,k+rT,
'..-..
<
1
.
,
JU.kfV'^
= ilVi.r +
-
Tj,[r,6/f,]
.
,
'
1
indx,-,]
IsL J. iriJ-rn]
J
T
.
.
.- lii.l,k+1
i.m
:
tTi
k =
'K,
n-ir,
and
:
1
ir.
i
Siiwtracting LUiterTi (4 35) from
!h
[lii,[r,,j,rr.]-hQ(rQ]li',Jr,d,m]~P[r,^jjU,
C'[r,,ji]"-
£a^-V*T
.
—n
I
li,)
*'
,
'
,
li
'/
.1
,
W-1-T .^.k-Tv ~
i
J
=
U.i
i /!"i ]
=
wj^i
iTf
[m J .rn
1
'i
,
[I'lO
xv j "
ifi
,rri j
3
= ^1 [nd .m
,
T
11'
~
bi,t if'O
/n ]
}v,i
,m ]
.
[ri
,
"
j
.
W
j
.
3
(ri<i
-TIT
n-,-1
;3 4) Uicl^s the fu!]ovv'ir'.9:
" f'it^inojli.uN.r.i-l
iiL)>+i' i^j>>i' =
..
iri-3 .n'l ]
.
Ej-V-M'
- i
-
i-
> Li ^-
;.Lii.tc;r;
.r.-i
1
*^Ai.j..'i..i--i.
]•
= Ci.lrid.m).
,
iil^^n -lii^y.
AT5^,,r..
.^,.n'i'.
= i^a^,,,,-'",
...
,
ij::^>.*r-'rr.
. l.i i
!
K =
.^
I
lili
.
t
t
-he
i
1
1
1
c '.11
I
to.
1
1
w
1.
"^
-'
>'
y
(
Ci
I
w
^
i
C'
'lii,M,r.-i]~f''-[aojil.i,[rid,m]"r'[nd]iil.n-i-1 " -'''-indjli J.fnd/fi-l
'^•^indj'"'
f-r 'p.
1
.
-Vi.
.
Im.
*''-r
'-p
...
1,.
,
A
such
1
.1
_
~
*
'
t;-,G*
'p
4.r'.*r
t"
= L),[r.d
/•-,),
'\,"-- i..'
,
.
]'
I.
J.:
.'
1
TC\
,
Thui, thi Frechbt darivstive, L^Civ
,,h),
or b^iiryf,) is defifit J e-
such that
-1
'•L'
K'
r,-,-
l.'-o jj
'
I
rid
U
r.-i
[flC(ndll
,ridj
'*'L
=
J/'
V.rri
,-1
[_hC{nc!]j
L*rr.
h^lf'^lj
P[ril
[hC[r.d]]
[hF.lr.d)]
=
Lirn
Ur'l rr,^
1
,
.
jj.k+i -
for
••"
1
^
I
r
1
^
1
I
I
mr.
k =
and
i
1
',..,1.-.
TT
i r.
i
.....;
C
TT
I
n
I
I
rr.'y
nU
^ .-.-.'
TT
I
U
Jr
z.i
.
U
1.
+1-..-,
Hi
MIC U
I
fC
',^.-,k-
I
I
I
r
/-,
.-
C 11 L C U
' *
^
1. 1
I
I
,-
C
C^A-^^.i
^. .-i
CL MC UT
,-i
I
'.
I
I
I
i
i
V
-i
U
»
'.
1
i
V
•
.-
-.
CO
.-.f
'J I
l"
,',,,
U.^.
VJ^
..'
j-|
^
(i=i,2) to obtein the Lipschitz condition in (4.35): that is,
where
r
f
,
\
...
- ^1
,rri
"
./
(
...
= 1^1 ,m
Wn^'^
"
'
i^2,m " ^'nn^'
J
'
J^2/ri
iJ
.^''l
Qrnd
rn
.m" -i
t:
,r."i
'^
1,m-^' 2;
,rri
r
.-I
.1
1
n\
-
p.
. .j'an'l- .t+1
- i2.k+i
y*v - i2,k+-i
11
and
Via the hypothesis that the second partial derivatives of F.vvith respeii
tc V ar.d z^ are continjous, v/e can Infer that the first partial derivstives of
F vvith
Ui;
respect to v snc
c €..10^0 pUo;Llvc
1''^-' l-'-wi
1' '11
.•''''fin'-
are Lipschitz ccntinuous vvlth respect tc v and
z<j
'«L>i
ic '.un
lo
- ^
^
r,^
V--I,~J, ^^u'!
i.:iC>
and
for
U
1
i r,
k = Ku\\
m
I
!
.
, 1 -
:M
.
_,
j,
_
-11''
i
,'
1
'
^
.'.
il^h^i^Xh^ - U(i;s,h)!!
,
i
If
1
fT,
lrr.3^:!:•:.a..:;.,!!^, ,,, rr.a.>...r.,
: ir-ftv't
i-\
_
-
I'
II
,
_
,,
'l ''^1
.r.
'
II
_
...
^2. i.^i
,,.
i,
I-
...
w
.'
-1
-IT"'.
.
.
.
^
r-
^2.hi--
-
,
rr^a::!!:ir„(ji,^h) "-f m^Ji'2Vll^
_
1
,m
ri
''
II
,
-
--n
- ^11
,...,
,
i--'
'
1
1
-''-'.vf'l
1
2,m''-
J^>^^IL IT >*' - ]->*!!
ia;.{max{!;.Ji .^*^ -
<
i
-
.-I
If
M A'
' 2,m''' 'I'
_
-1
-.-.:,...\-f.fli... fi;
- i--.t.f!u.-.;.Hi.
'1
1
1
^ r,
k = rnr,
um
.
ni)
^v. ,,^!!,j..'W,^^,., - jr. „.,!:::iur,k=rn:-.,I<ru.ni!
I...
11
Ji2.h''
z<j
We
haVfe riov; tet. th'.ngi up to invoke
th;:. tO'.nt
we
summanzed
sti'td
arid
i^ro'-.'t-l
Ifemma i
vanoui
from [kelTS] To arn.c gI
]
These results are
lernrra":..
as follovvs
Theorem
4.1
Consider noriiinear, decouD'ed systems cf different! si /algebraic
eC'.iaf.ons of the fcrrn (3 1)
Let
- TIT
- 'mT
De a unique local analytical solution to
;i c-^;. Li.;2. i-La'' "-Xa' "'"-•
nuDOthesis of lemnnas 4
TI
:.
.
..
.
decoupled
C3
1)
,_
,ici. v^..
rr
r-
r
-
1
- -
'
1
'"-iLd- "-=<j»
4
,
; >-.-
-
T .
2,
-
;
4
*
r-i-rr OiO'^i. iifipiiLi>.
indei^
are staple
i^Ib ul
4
3,
-
»•
(3.
-
1
lu.i
4, 4.5,.
* * -
Suppose the function £in (3
)
r
I
u..r>
On
l.'.o,Afin'-
^^rr-^^^
<-»'C
and 4 6 are true.
•-,- .-
Liifc-^cp Jit
*
I-
-
-«
-
(.liw^ja
i'
'..
, - .^
r- ^ - ^ ^
i.^.l Zt.'\..it'j
<
* \.L>
one systen'.s of differential/aljepraic eQuations
G
1}
^
-
^
» ;
..
——
n.^ni itlcdr,
of tne forn";
5.
FSFF Convergence Analysis for Smooth Index One
Systems
r.ovv
Vv'e VYii:
of Differential/Algebraic Equations
utii-zt the results o&t5ir.eu in the previoui
suDsectioni :o
prcve cunvtrcencc' cf our FSFF block implicit or.5-£twp mcincui
to smooth, nohlihear, decoupled
equations of the form
Theorem
stated
m
2 2
from
theorem 2
Theorem
one systems of different'.al/slgep.'aic
oDtcin the classical convergence result - vvhich v/ci
thot
1
5) applit-
Essentially, vve have set things ud to invoke
(3.1).
[4; to
inue;<:
(l
is.
5.1
Consider nonlinear, decoupled index one systems of differential/
alqePraic
eq;.,Gtior:s of
the form (31), for v/hich the first and secor.u
derivatives exist and are continuous. Let
be a uniq^^e local analytical solution to (3.1)
theoremiS 3.1 and
~-
Then ior some
»-,
•_
.'.•
_ •
^
1
Suppose the hypothesis of
are satisfied.
pr, >
0,
and
v^,
=
•]/
the problem
has a unique solution
solution of
v/^,
e Sp;j(\y\X v«'hich converges to the analytical
L
I
I
-^
'.
I
.'
r.-O
~
Ali-C,
tl"ic
th£l
IS,
Itvi. ih
crdtr of
th'.s
convergence
is
there e::i2ts a positive ccr.st£r.t M„n. such that
•"'h''
-
'
-con"
•'-'
References
[1]
Atkinson,
K.E.,
New
Sons,
An iniruuuctwn
to f\'umcncsJ Anslysis, John Wiley and
York, (1978).
12]
Frank, R
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II
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