Convergence analysis of block implicit one-step methods for solving differential/algebraic equations

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iiiBiu^iass)

₅

MIT LIBRARIES HD28 3 TOflO

D0?3nba

fl .M414 r\o.

wv

MSA

WORKING

PAPER

ALFRED

P.

SLOAN

SCHOOL

OF

MANAGEMENT

Convergence

Analysis

of

Block

Implicit

One-Step

Methods

for

Solving

Differential/Algebraic

Equations

Iris Marie

Mack,

Ph. D.

October 1991

WP

3349-91

-MSA

MASSACHUSETTS

INSTITUTE

OF

TECHNOLOGY

50

MEMORIAL

DRIVE

Convergence

Analysis

of

Block

Implicit

One-Step

Methods

for

Solving

Differential/Algebraic

Equations

Iris Marie

Mack,

Ph. D.

October 1991

WP

3349-91

-MSA

Oowey

M.I.T. LIBRARIES

DEC

2 1991

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 in [5]. Of central importance in this theoretical

discussion is 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 - _and, if

so,

what

order in h this convergence follows.

The

first section of our paper will contain a derivation of our FSFF

miethod and

some

theoretical tools necessary 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

2

consistent. In section four

we

wil] prove that our FSFF block inriplicit

methods

are stable. Finally, in section five

we

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

ii(>^o) =iJD such that X e (Xo.Xf

_J,

tI^O,>'fir>WR", and f:lXo,Xf,rJ X

R^-^R"

is {C^, Lip) on the region [Xo^XfJ X

_R

.

We

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

v(Xk+r) - v(Xk) - 1 l(x,i^(x)) dx = . (1.2)

•Vi ^k+r

We

are particularly interested in the class of FSFF block implicit

one-step

methods

analyzed by H. A.

Watts

in [7].

Watts

considers the grid points

{X, I Xj .= Xo

+ _{jh, rh e (0,}

Xfin - xo). j = 0, 1, 2, ... _], (1.3)

and generates the sequence iij], such that ij approximates ^(Xj) (the

4

implicit one-step

method

as a procedure which,

when

applied to a single

equation of the

form

(11), yields r additional values \^^\, yi<;+2, -, yk+r

simultaneously at each stage of the application of the

method

where

k =mr,

for

m

= 0, 1, 2, ...

.

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+>

(i.4)

such that

and 1 < i,j < r.

Suppose for notational convenience

we

let

-4 := la,j],

I- _

-,-:=i *'iH

n: = . 1,2,

1 1 di

Therefore, equsticns (1.4} can be CDnclsely vvritten 2S

-1

Assuming

thit *^ e;;i£ts, Vve will further simplify

system

(1.5; tc

oDtam

Sir,, = hEZrn * iyk * 'll^t- ₍1 r

such that.

G :=

a'

i,

and

-1

^-A

r:=[l. 1 1]T.

Utilizing the Kronecker product notation, FSFF block irriplicit one-step

methods

may

be applied to a nonlinear, decoupled

system

of

differential/alaebraic equations of the

form

B><.^^).Lj''"))

Fd(x,v,AA),ii^(x),Zd(x))

fi

suDiect to the initial conditions

r [vn VV..1 y Pn y pn -^Pn :_i" 0»"lll|J • •» .... p. Fd:lXo,Xt-in] X RnO X R"* X Rn<» ->Rn<« L: [XO,Xfin) X Rn<« X R"**

^Rn«

iM

:= Iyd^(x),i^T(x)]T^ n := nd + na,

and the Jacotnan matrix [^F^/azj] is of

maximal

rank on (xaXfJ.

If

£

is twice continuously differentiable, then it can be proved, via

generalized Picard-Lindelof theory 15], that the solutions to

systems

of the

form (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 := _\'ik*\'^

, Zk+2^. -. ii'k+r^F, Zo := _U'O = _\Zd\

_Id\

-, Zd^F, k = mr, and

m=

1.2,3.I —I ^ I

Let us utilize the Kronecker product notation to define the following matrices: P := _^r X In, Q := B X In, R := Dr X In, such that Or 0.

8

ij„. - Pij~..i + hlri7:~ + Psr~..i1

M

Q'l

The

second section of our paper will contain

some

theoretical tools

necessary to prove the convergence of FSFF block nnethods. In section three

we

will prove that our FSFF block implicit

methods

are consistent. In

section four

we

will prove that our FSFF block implicit

methods

are stable.

Finally, in section five

we

will use the fact that consistency and stability imply convergence.

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 is the question of convergence.

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 + Stability

^

Convergence."

In [4] H. B. Keller presents e general abstract study of

methods

for

approximating the solution of nonlinear problems in a Banach space setting

Hisbasicresultisasfollows:

If

8

nonlinedrprotiJem

has a

uniQuelocal

dnalytlcdlsolution,

and

Its consistent approximating

protlem has a

stahlP

LIpschlizcontinuous linearization, then theapproximating

protlem has

a

stat'le solution

which

Isclose to the analyticalsolution.

Keller considers the problem

G(v) = 0, (2.1)

such that

2

flnd Fi. _{(i = 1^2) Qr8 Qppropriete Rfin^ch spscss.} On ₆ femily _of B9n5ch _spaces,

[b]^', ^2^'}, the family of approximating problems,

Gh(Vh) = 0, (2.2)

where

GhiB,"^ -^ 62''', and

< h < _ho,

are also considered in [4].

To

relate problems (2.1) and (2.2), Keller requires that there exist a

family of linear mappings {P^^, p2^}

where

P/'B,

^

B,h, _(2,3)

and

lirnllP^MI zllvll, (2.4)

h-»0

for all V e B, (i = 1,2). For notational convenience the following is defined

in [4]:

Iv], := P,hv, (2.5)

such that I_vV, e B/'if v e B^ (i = 1,2).

\ \

Before

we

define consistency, stability, and convergence,

we

will define

a sphere about an arbitrary point in a Banach space B as follows:

Definition 2.1: The sphere of radius p, about u e B, is defined as

follOYY'S:

Sp(u) := {v: V 6 B, Ilvul! <

Now we

are prepared to state the definitions of stability, consistency,

and convergence, to be utilized in our analysis.

Definition

2.2 (4): The family {6^0} is

stable

for u e B, if and only if

for

some

ho > 0, p > 0, and

some

constant Mp > 0, independent of h,

llv, - Whil < _t-VllGh(Vh) - _Gh(Wh)IL (2.7)

for all

he

_(0,ho], and all v^, w^., e _Sp([ulh).D

Definition

2.3 [4]: The fanriily {6^0} is

consistent

of order

P I 1 with

GO

on Sp(u) if and only if

llTh(v)|l := _IIGhdvy - [G(v)lhll < M(v)hP, _(2.8)

for all V e 5p(u), and

some

bounded functional n(v) > independent of h.n

Suppose u is a unique local analytical solution to

system

(2.1). If

we

substitute u for v in (2.6), vv'e obtain the following;

l!th(u)ll := llGh((ulh)ll < ri(u)hP.D (2.8')

The

significance of definitions 2.2 and 2.3 derives

from

the following

theorem

from

(4).

Theorem

2.1: Let 6(u) = 0, and Gh(Wh) = 0,

fnr =nrriP w. e 9 (iuu o > and alt n e (0 n-i Let \bLi)\ dh ^teD'e fn*- u

and consistenl of order p i 1 vvilh

GO

at u.

Then

IIIuJH - Wf,ii i n^ndOhP, (2 9)

fOf Mj, •;

D

llfiliTirn ih&nrarr "'

1 _v'^ '.A.-ill fn^msnii rlpfinp _{COP'-'erO'=""!'"P *''''d} ^^"'^ _QT''''^'

of convergence as fcilows:

Definition 2.4

A

sequence of nurriencal appfoximants VV|., e 5p([u]|.,) is

sale to

convsrgB

to 5 unique local analytical solution u of (1.5) if

lim i![u]h - vv'fj! -" (2.10£)

h -^0

for p > 0, and for ail h e (OjIq] The

order

of convergence

of this

sequence is

_p

if there e;;ists

some

bounded functional M(u) : 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; if there exists a bounded, linear operator

L(v):5i

^

D2. such that

1i ^\-\

'——

—

ii,..ir

—

~"'""

= *-'' lv.'i:-*0 'I"'!' _C^ 1 ''1

tzr ill v: € B, Ihif. L(v) is the

Frechet

derivaiive

zi H al ': -h 5,.HI

vv^e chcll see that fcr our differehce approiamitioni, oDtained via

appl-nU-g :.v.;" FSFF btCC.K methcdi. to i.j£terni. cf tht forrn vl 7), consiiter.cu "li dtter'Ti'ifiec Dy iirnple Taylor

Scnes

copansiotvi. Bccaus8 the staP'lity

venf'Ccticn for general nonlinear proDlems of the

form

U

7) is n:i a

"itandir:!" prccec^re,

we

will proceed as in [4] in particular, Y.e v.ill

I t'U U wC U'.iI w*>.Ck-1 f Iv\4 'Z'*.Uwy ^.V L11C'. w>1 u I111C'SI Ik C U ^'I Ck.>ICIfI^ <311w V11C11 QK^''^ ^1 1v

loll 'vv1r.Q 11 rr fna ar,d th e c r ern fr on'l 14j

Lemma

2.1:

l't . '.l!r .3i.i!:y •.•: ;i,ji.p!r-y- Of...;. li-wvtf rrtvIlT'. uti iv>_'.1vc:• i;_|.,'..v|.|.i .•!!

'.., iw^|V.

Vf.;. AC.'z L.ii :w. lulLj uu....j^- ili-.c; ceo C^ l::e ^c.;^c. - '-: I.ic

Spheres: that li, :cr

some

Kq > 0,

i:: -ir^^ m; _\\. i': \^\

i.-n *"n''' - '-LI- ^— •—•

'..II, _{t;_p, v'f^i}'' C. b Ui:!1.i Mil^ _.(.c-UUi.- wUiUiriwUvb Ull ^fQVvv^.. :.ii2i. .w, iU:

... . .[_ . . J

,i_r,..'h-' ^i-i-.t'fi.'ii ...L,,vn *'ri"' • •

\

It v.'t, = i-,rj !Ul

cC-Zwit

;_],i.ric.i .uS i C.i.I1

'.{ iC^/. .. .; :.'__- i..

Theorem

2.2:

cor:.'ste;-,t :: zroir p . 1 vvitt-,

GO

at u. Lei Ihe nypolhciis. U) and (n) of _=...iu ^.1 .i^._ ,. .Lii vvf., -

iJ.ri-ThrH, Tor _pq ; and h^, : sufficiently small, and for each h s (C, _{In:,!, if} pro^:l6n-i

Gh^v

= r-.u:

The .nequclU^^ in (2 ;5) \i eqi..;v3leni to the foilovving.

"Orui-" 3f Ccn/er'sncs - Grdsr of Conslstcncu."

A

'jo,

we

have convercience if p =

_p

2 1.

Ndw

let 'di 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. .'e v.-l^ define _X,, '''- _{t'e the}

local discretizdtion

error

at the rntn step that is,

T

-

ff T _t T _t "^^"^

(:.i£)

1=1

n"i = 1, 2, ,

t ,T -TIT _ r T ,. T - T ^ T

n

'9- '

—

' ~ '*d *i ' isd =3 ' ••

lo c U'.i.^^-r' .L^.c! c.i.2.y.:.-:, w.';l. v.,L!fi lJ ..ri^ iiw^.;G; vcIl.*; prcjietfi ; i . ; ; lj

. .iC dU.li^-. CI.. i._i, i_'i, us.u L^i 1 cTc; ^C _^.^^ '. . _ 1 _ I . . 2i ;.!;= flJIB/floJ

locdJ

truncation

errors.

Definition

2.7. ^r.r

overall

order

_3C^rd*r ^^^ ^-^•' ^-^' "Icck :r:.?]:c:t

Gr.c'~-:''.c> IT,ith CJS \\'-3] IS uStWiid 8i order - ' -^' ' -^I• "^r-.• ._- . -.' !.-•-. tl-.a* l».: = «^, C, r V -K-^1 + ri.'hK!*M -J..ir (w = , 1.2. It-.' 111 rnpfc.'C11^j.f1 *^ *-•I t I J _11-J- II I I-.•_VC11 .-11Ci !

V

_ - * -. 11cI 111-:

-\-^^-i,

t:;e cvi:--:]

orj?r-X,,_

of ju- r^r

UI.C :-.t_^ 111-.I

3.

F3FF

Index

One Consistency

Analysis

for

Smooth

Systenris of

Differential/Algebraic Equations

i ::= :::2i. j:E, -^ 5^, difinad ^n (2.1\ ccrrtspcr.di

U

1-1i'lYI ; -dV Fil'V '.' .1:.'I <<•' l"VI'l

—

* •—3 •.i*^ •••• = I' I~ Ir /-. 1-. *i- 5 - p !v.

.

' .. _ _F,,.J M TIT ;-• u- Tin- ^ u [v. •.. . !

—

• pnj -> fv. •..•. 1

—

pno ^i'""!.!.-' •Tin' f^ ... ._ r,,T ^ TIT Ti-iC, P^••s;-•» C;^;

-1 '- ;ts 0' t'-.e

norm

c

te define:: in (-.2;, snd tf£

--V .fc-.1' I'lri"*' '* *

*' ''- taj.ij... .^o*- ^.^ ... -. -.1- .._.,.! ._ _

definsd '.r. {2.2). and the set of functions

'.' -tv. V. 1

—

Dnd V Dni v Dnd

which

6''e c" c^'^ss i"'' on ••o.Mf,r,].

We

will denote this B^n^ch space

n

P'([vn.^-tini f^^" - R"^ •• R'"'-- The elen^ients of B, are of the

form

- l^c] ' -143] • i- ' ' c-uLIi '.lie'. 1':'r ,1' ','r ;I I -H3J - I

-nd

• -ri' ^dj'*-'"'.!•'Tin' ' \.. ,!v- v.. 1

-

Pr.j -Ha]•'•"•'-* ••••tin' "^ i»--ij.•yip,J l^ .• .'v. v.. 1 -k pr^d •^1lu _ 1 -^

!!

£

= Bi (i = 1,2), then

we

will let the

norm

of v be defined a;

II::''l := suodlv!! !!rii }

:=supltii4dlli„J!y^.ll!„.yi

sup{{ su? iy.(K)!.. sup !v^ (;)!.. -ud |r,(:01} : x e [xo^k^.^,]

if _i; is a 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

mgnner

in vvhich h is defined, v/e see that

K

tin

For notational convenience let us define the following r-nd-dimensiunsl

arrays: Uind,rf,]:= [iiJ,k+1 "''.•••. ild >:+r'''

V

, Uind,0V=lzj/. -.id/]^-2(nc.m] " iSid>;+1^. •, id>+r^r. Zjndo]

=

'=^,0''". _{-. ixi.o'^]''''} ror

Also, let u: utilize the KronecKer" product notation to define the

foUovving n-iatrice:., which will be utilized to extrect the equations

correiDonding to the differentia] variaDles fron"! the difference equation?

denned

in M,9):

and

P[r,d] = !^'r X lr,d.

Where

l'\, B, and _[v are defined as in equation (1.8).

For each such grid of the

form

(3.3),

we

can define the following difference

schenrie:

[nC[pH;]"' _{iU-nd.^l'^NiyU-l} -t''C[nd;2(nd,m]"'P[nd;Zind,m-1_V^ - _iV-nd, I • il •Is'

ija^'-kri.. io,»:+) ;i<i,K+i' -sdl--*-!- ~ Ud,k+i' •- -iVaO^k+i^ id,k+i' liU.k-^r - Ua.k-ri'

C' .1 - I"!'h^'m,Ti'Qp -Tt-P, ,/ip p Trrir ji+ Pr j"'**!

'._-'.•-tL .

1 :• rr-, .:

M

-•--'•,

\

1 1 i < r.

The

Bansch spaces B,^' (i = 1,2) consist of the

norm

to be

(R^'- ;; R-^ >; R"-:^, The elemsnts cf B,^ arc ci h^ fen-;; sucn tnat ',' - ['•. ,,T „', JIT ... . _ r,, ,, _]T M-i.Oi " '''.'.1 _'"J.nd* ' .., , _

'

.. rr

MT

,3]

-

'''S.nd+I • •• '^s.ri' > V, € R^. V, ^ R'-, I ,i.

!: _ij., e P,*^ (1 = 1 _.2), then the

norm

cf r^ will be -efined a; follov^-s;

!l ;.. |! _ .r.siv^: 1,1 )i... .11 II.. II 1 .: : V.l

i"na,-,-i (Tia.-; iv<. ,i,

max

v, i \. m5.^ ivr lii 1 i s ^ 1^

'

11 in; " • _u<if]i .<iri "

' i:,1 i-;d

.J.3

!'" _y

is a sOiu'-icip 10 1.2 1) then 1% = v_.- r t'of 1 i s i !i<, ano <.: 5a.' reciute': ii

t ' .. .. .i ' , I . ., ( I < . ».1 -T S

lij^ni .= ii'ia.',i.|_v^i! __ . _ii^.

.j tI' .. I i. 3 1 A.I- \.'-j:

Tne rr;a:i:-!nq 6.?,'

—

z-j' •:i<i!l]f\i?.Q in (2 1), and applied to

: 1.r I..iCu: ihu^i^11,^c r

, .Lj; U; ..| - I ,—.'

. tJl i Sepuili_ vu -1Ic riiu^,;;l;^ i ,

C'^LI> \.1nj'. Gi iU Gw' 'iVI iI L'C OiiwiVn 1. |i^K . i; _ 11 . ir I .fII III 'f M - li__II '.: .• ; !f V. --.-1,,*

J.:_.l Li^ ._!., l.llCllV.-_v_^, C.i>J V-.U-. .CUUwb^ lb

Alio, bccsuic Vv'c ii'c Vv'crl..]n;4 \iitl'i srnooth func-t^ons, vvc f,i";OW inat ci- the

gria u5fTr,£u :.y (3 3)

Decomej

denser, the

meximum

value in (3 4) vvi]]

3!

Proof

of

Consistency

Nov.' vye are equipped to prove that cur FSFF tlock implicit one-step

rriethodi (1 9). applied to stricoth, nonlinesr incex cne iyite.'Tis of the fcrrr,

(1.7), ire CQr.s^£t£nt with

_GO

£t the soluticnj: := [^.z^'^V - ss defined in

Gefinition 2 3. Using notation similar to that utilized in ,2.6 ), Yve Ojtcih

the fc'lovving

=

[^^

-)T. _r..{ .--iT r„(-)T]T^ fT ,":1

.''.«'.11 ^<IW-.

I [hC[nal^ I

li[r„3,rn]-F [T,iUr,T)-h!D[r,v2fnd .rp]+R[rri'Zbd .m-l ]!

1 I

i

^

'ic C2.16;.

whe-e

1-. — Mil .

S'-,l-!

1 < 1

<r

Taking r,r:r-rr-,s of ticth sides of (3 S)

yieldi-such that

r.

Qirid.rr] " ^i*j!.••

•,., TIT

Uirii.rfij - ii4.3,i-t-l ' •• Ui>-i-r

Vv't ca;'. riOv,' sj:7"irr;anze our .^SFF ir.de;; one consit-ttncy resjlts as

TLi11.. r,

z-.

Theorem

3.1

wuiic-iuti n.niiicaf . .jcivi-ui-iitu sy^'-ciiic- oi ui iti oilia.* c.hCi.-^ g.-.

t-^uations of the tirn'i ( 17), which satisfy the aly6lrc'iC and icca.

SiTiOOthr.rSi- DTD'-BrtltS dliCUSS^d L*t

- c unique ^cia! analytical solution to (1.7) - _ui of classT'-*' or.

::.o.:'fir.'

noru^near roo* finc;ir,9 prcDiem so accui'ate'y, ii.it^ ihz'. there e;.*;::: pDZKive

constants c^ and r^, sush that

I'n- 4 -'i _r t-''*i

'in',. -i" ^ f. n''*^

'ill.,3,M.J » - ^-a ,

I>.'! > . !II - .' i.

Then there e;ri£t£ an

M

> 0, indepenaent of h, such that

jU ,' ..Ml /

M

l-,r

1

4.

FSFF

Index

One

Stability

Analysis

for

Smooth

Systems

of

Differential/Algebraic Equations

Ncvv vve will prove sta&ility of cur FSFF block: implicit, one-step

methodi

(1.5) cpplien to nonlinear, Gecouplec indej; one

systems

of

c;i:ferential/sly6l:ra;c equations of the

form

(3.1). Cur stajilULj analysis

c;n be

summarized

as follovvs: The family c: mappings Gr,;5,'"'

^

b2-\. define:

in equations (3 4), will te ccnsidered. It v/ill be shov^n that

_IyMiW^

-=•=

Frechet derivative of S^, at L'l,, exists on SpoCUi,?,

where

x

:= _[jJ,LyV ^: £

satisfies (2.12) and (2 13) - the hLioothesis of Lemima 2.'!.

To verfy

(2.12;, it will be

shewn

that if _'ll}n i^ the ".niqus solution to

UCl^lhMATinzij^'n. (4.;j

(v'her't \il]fi end '^ij^, will oe defined shortly), arc if the'"e ei^ists a

constant

M

^ C. such that

:i!//;^.!i inlllj'i^/!, (4.2;

then

that IS, _L^C^lr, '-' IS uniformly bounded at the center of the sphere

_S^^'UV'

for- all

h€

(:,*!.

The secono nuoothesis of

Lemma

2.1 - (2.13) - will be

snown

to De

satisfied if it is

assumed

that the function F has continuous, and bounoeo

v/t

in

ncv. tq;;ip:.e^ Ic provt itGu-lUy o: tl'ie FSFF oloci: itniiiicn or,r

Gil it:"en*iai/Giuctr5iC eqi.ciions of Ine forr;", (3 1)

4.1

Derivation

of the

Frechet Derivative

Cir-r-:^ '•? -•-•-^-'a^-

'

- ^- ''.'''" -

_"K

_*h'if

/ "I

(II .^- lit'' I

s,-..::

Tnc Victor _ii'h Silisfis: thr :cillcv;ing difference scheme:

'''•^[nd]'"' 'llqnd.mr^'-^inijiy. ipd,nn]"^'[rdjli:!fr-1"-"'^'(ndllii. (nd.m-1)-*IIind.m)=ir.nd'

r

'...-

^ _ .- ''1 /'^ Y/here - •.. _T .,• _.1 „• T'T L«, - - '•...' T ',. ~"T l54nd,mj - '.^.k+i , . jjjj.fr+r ' ' tr. , , _ r T ... TIT.

is- ir.o.mj

-

'114-1-1 •• ' ILft-r •

r, , - ''" T r T]T Mna.rti) ~ '—d,K+l ' • ' sJ.k+r < '

_ . r.. - .. TIT

a,..',m - lUv,'>-r1 •• iiv>.Tr ''

I ^ IIt — ^ fc',

k = rrif.

1 . 1 - r

Ttirz li c rcciJu3l term, rrn^.m)' 'f' ^^-^ because _{jih does} not satisfy the

Cifference scheirie (3 4) e;;actlii

S«t:raiting sjsten'i i4₄₎ fro;t^ systetTi (3 4) yields the follovvinq;

if'^'inci'" ^l^nd.m] " '''^indjiiind.m] " Pindjilm-I " ^'"indjli (nd,m-1jJ _=Lirid/rtl..

F rv ; 1 - r .'• .•„ ^ _ '-,

such that

!'- - 1'- 1 -

t.>-i-H'-Mj.mj - iiirio.m] aqrid.m].

ii[rij,rn] ~ Idiraju] ~ !iL[uij\i],

ilrr -- _{liirn^. li'hd.,nr}F,

liIrid/fij = tind,m] ~ iJ:^_ind,m]

,

/•n •- 'At >

—lUv KtI = iir.+i ~ Uv; >+i.

1.

h". .

_p.

U.5)

]

n

- r

The Frechet derivative, _Udx'lhl of the mapping G^,([rlr,) is obtained tq

r\j ,-1•-•-, ''-.it

«

._,'-k = rr.r, 'LiIW 1 11 r.

..C-. 1..;- '.iMiii-f •.litr- 1 ii^.i'. (luUw oluC !-1 V'T-.'/ (.0 ii.Ot-.C (.tit uil-j'.. 11..j

ucfiniriOfi.

Definition

4.1 The

Frechet

derivstive. UCIiI,,), cf G^,(Iry cr^

C. r' •1 ^ i- ,--.* \r,c.-A *:

-"HQ-'i-'fT -^ LsiUitJ u-5

I .•f..-.l •, _.-•,=.,,(/ ,'I..i , /'^^'•'•l |1 l4~! c11r-. *'•,i ^'^L-1I (.II'w '-?l,.. = u-r-' •! 1 I vM It f'*• i^'-JJ ! ;rA, .r„; Fi^r' ,.! r_i-,-. ,1 ro

t*rr: ---1 [.•«- ..i-,J "^ L' -J ]r^[r4\^^''[r4 Urnrri'j J'+i = •Jad k+1 Jad k+i k+-. 1^+i •-Or,a,r..i for 1 <

m

i n,. k = mr, and 1 1 i . r.

Also, rec!!'' thrt E''' is the bsclv'-'crd ihifl operator that is,

for I i

m

. H, and //n =

n

arid

4.2.

Proof

of

Fundamental

Inequality

For notG-.ioric; corivenience, vve vvill refer to the inequality detinec in

(4.2) ge. our

Fundamente}

InsqusJity

(F.I ) By proving that our F ! li true, Vve Y/ill also have satisfied (2.12) of

lemma

2.1. Utlizmg (46), v/e ;•

that si^Ltem (4.6) can be revvntten in bloc^: matrix notation as fOilOvvs

^-TTiii-m - %r\' iMVii that Rrn !fX[!r,^n,d,:.ij I

m

%u

= L,.;, /^r,-,.i + i'm.h -- n ',

P:r,^|li--r^":[r..:!_{U:[r,c),m-1 ]*•-:•--}Onu/r,;

"Dr.

P[r::] ! hP[-::

-rr.

u:v,- \:i vviil utnize Elccr; Gaussian E;Mriina::on to solve susterc. ;4 ;; f-r

iljnd-'na.m] - > ili'nd.m) ii[r,3,rri] ''.•

Yi'e Vvi'.1 sCiVe the fcllO'vviriQ n"iatnx ecjotion

"'"II - 7" 1' . ' rr'tL('-id+ri5,rrf] - ' m-1 ilind-^nj.m-^] * '•""• •^'ind]C(nd,m]"*' '^'ind)"^[m,dd]l'' '^Q^.hd.m] * _{t'nr.i] i-^in-i-l .dd]''} ' -Q^.lnd.m-l l'""^' -!j£_.ln3,rf.l'i ' '.4.10; w' '^•--1I '.ft•J •-Ir,-, lrr..'~f'•I''r^^^l*•. •"':' /fl -hflr i/yr.. ..1 /y(T,,ijl / n-;-l ;= />.^1i'^.^%..^•/ftr,-\ •-'I Crni..rT;d hP[r..i]//:---1,d.5] Orna

>^. . _ _'-i-'5-fr» i-t-n-iri i+n ri k+ri-iri t+rii

'' _irn.ccj ~ r".*dd ' ^-od *' ' '.»dd ' ^"60 ''•

im.da] ~ "?"Jdd ' '""da *' • ' 'Jod ' ^ ds "• //• - - "•=-' ' >*i I l+r"

-.3d "

im,.33)

—

s'-aa '

•

~a3 ''

"'

' im,dd J,

M,

. = P'-.r.dx^-nc?

''_iro.QGj - ^^

yV' -' _e P''.ri.axr-.rMJ • • IIM.i'JJ • •

"

[n-i.as] "^ ' ?Vr j^' e Pf.ndxr.nd • * [m.ddj ^ "^ >

P-

.' - _X I

-Ilv:_.[r,:,r,-)i = i-^%_,d>+l'''' • > ^Il^.dMr'''^^'

5ind 1 _i

m

< n. V/e vvili then utilize the solution to (4.:0) to obtain ir[ri^„,j.

that IS.

li'lnd,--;] =^[rn,dd]li{nd,rMl * >^Im,dajii{r,3,mri-'^[m,ddir' •^Q:^(nd,m].

'AW)

for 1 . rn i n.

Becauie _^J'^^ \i nonsingular,

system

(4.10) can be solved fcrli^'r,^^] as

folios

such thst

for 1 : i i r, k =

nr

and 1 i rn i :M.

Subsystem

(4 12) is then sutstitule: in^

(4 10) to oDtain the following:

= _^^[Pdl * _{^^(ndl ''^im-Kddl "^[m-1..dsl^-^Im-l}

..aa)!'' -^[rn-l .adl- ii(nd.m-1]

;.ucti that

rL-,'..u.- = ,C'[r,^]r{^H,rr.]

w(r,ijj n-'* im.cd]' "—Iir_.[nd rn]

"*'

"[m.ds) ^"[m.sa]'

—

Il-_.[n3.m]J

^ _"indj "• ' im-1 ,Qo]'

—

Il__,[no,m-l]*^[m-1 .da] '^tm-1 .aa]-

—

^;^:_,ln^,'n-1

)

i^1 « lit ^ J I'

i i'lhui vve ottain the follGVv'.ng recurrence relationslv.p:

iuch thct

J .^... ._ _...r-^rj •• _{-incj '''imoQj ''} t^..oa) ''' Im.aaJ*

"IrD.ao;'-^ _incj • _''indj

••'' im-l ,dd] ~ '' (m-1 .oa) '''(m-1,aai' ''[m-l ,adl'>'

in''" - •'

-'rr,d '• '-;nd> '"trn.QOj

"

iTi.da) i"lrri,aa]' "im.adj-- Drri''-'

ff,.- ' - '•,-, . 'vr

!'1J, ,' |i'»'^

''f-

M

I'll, ,'i + i'V '(•"I'l '.A

'^':

IW-. . _.i._,w. . _: ;:j._.:u.iai wUi;v:.,ifii^^fc, Vis ..' ii i iC. li ,i i^^i !U iu. ., ,„:,::i.

*1,-,- 1-.+ .,r H.-«.>--. c ^3.-..3r-- -V ..,,,,v»•.-•,-•- '^' I ...l-'i^l-, c.i*i-»'--

*••-*c.iOVi "iTii^ cifT&rer.ce eQJ5*.ior;;

•

m

- .•• Tp...i...1 .nrl i_.ijT>•"•'" .m. .u.

lb; I _ .;. - r._, C:.- .

f, v.

Lemma

4.]

11 'dit ir,;lial cG:".dit.;ur,i. for (4 14) are coGClly satufieu - _thsi ;;.,

"1 . ,11 '..' C 1 '~'\ •_{'iiiriO.rri!''} - '''m- ^t i , . •?•-,--1 r-- - >'

n

Proof

of

Lemma

4.1

The inf.qijailtLi (4.17) vvill be proven by indu-ition ai foriCvVi. Eeiauic

Vr,- 0, a^o ll^r,^ ₀₎ "

L

(-^l^J i£ *"^£

Term

= 0. uO'.v suppose (4.17) is triis

for £G;"i",e

m

^ ?l Thei'i (4.15) and (4 15) impiy '''', ." . "^' ('(•'I' "1' .11 ₊ "V _I'f-V

•'iiinC.m-rl

J''

-'<>• _Tn-r\ ••<• _{•'iL(nC,mJ''} M:_Ln.|-l »H'lt

From

the theory cf firr.te difference equationz.,

we

knov/ that (4 16) can

be scivei fcr V^r, as fcllov/s: t •rfi = r- 1'' ! ' ' TT|i^i'' '' > I' ' '^ "*^-'h 'iI' 1TT''M-iih I ! '

-, J-i'' • '• ', 4_ i.=—- hiilir*" II- ,'

ii=1 i s=1 11=1 i I

m

- i,2S_iW..''ii ii" .1 -^ Ii^L:-'''"'" 11'' ' i" * •• * Mli".vi.J,r

m

Nc'vv vve

vnU

prove tf-:2t II''^^''''

'ii _and

i;;K^,';h;!l 2-e !;r;-^o^Tr,:', tounded 'or

I ^ I - 111, u nJ I ^ In 1 .> I

-Lemma

4.2

Suppose there exist positive constants l^.^, l^^. l^.^, I _^j, and L^., such tlial

'i-'df 'I - _'-i'i'

lll-'ij J ll - i _43, and

for 1 ;. ] i r, ar-ij r = mr. Finatly, suD&ose that

A

is smaii enouoh, sucn thit

hil. ),

Yvhere

h 5 (O.^j,

and

Oilij

IIV ''•,'''1

. »-| " l!^v'' H _(d?''

Proof

of

Lemma

4.2

Firs*

we

vnll tackle the inequality in (4.20): that is, let _{us oDtain 1%^^.}

Frc-iTi (4 14)

we

see that

where

'•2

=

_''^[nd] * •'

-'[nd] '^(rrrl.odj " ^[m-l.daji^im-l,ac.)^'' ^[rn-I,ad)'i''

for 1 i

m

.

n

Lf. iw.: .^;-.T 2 IL'ij.'. q. i.V| -..i- i.,i.. 10 '^'v;.i^j,f'.i

W-—

^y- Vvt r. ii. i:.',Y '.^".-21;; '_ fecund 'ora, vie relating the

norms

of the varioui matrices in cv, and to the

LipschTtz constants in our hiipothesis.

''-mo]" - I'- * 'nd'i - ''^'''

( rn«f'''f'l t+ii-IMIH I Mill' 1 / i . r ' - rv'

-

^d

-da

L«u the hypothesis,

''''im.ds]'' - "'-'•'^?''Jdd '^ '^da ' _'^ • •

^3d -d:,.

< ms'^'-' ' '+11-11'- < • 1 - r I _ r^.'-i

- ^ad'

iy :;-,£ I'J ^-o-1I•->.'I-* for \ -

m

. 'Y^

!«'JVv ViC i. !:! jT: iilr -.I.t !..;.W.Gil;. |.'^ , iiiliLfi YV 111 ij3 L..iiitcu in •--.;

.».-..-[;2a- -

h^R:

* ... . ;:, hJl

ji, .= ;^:;i/(i - _h?:-.

Ulilizinq Ih; individual bCLinds for the

vancus rorms

in c^,, (4.2T\ a^'id

< 1 -^ hn-;!,)1l

for 1 i

m

. n. The following

norms

vvill be utilized to complete (425)

Vp. ," _ ii/t Y I II _ 11,.) II -1

and

ii'^jnQj'' •" _''-r * _TiO'

"

I'-r'' ~ I'ii''

Thrrefore i;4.25) _{can be completed as follov/s}

..., i ₊ i: II-III f .'I ₊₁ / I ^ .• c,..r,'f- 1!-"' f I'l + '

/I

V ,_ |U IIjII ,n /Mpllt .I!•n I!!. •- TV /nr-'if 1u: .: t ..-,/7J.

Therefore (422) can te completed a;- I'-'i-v.-iWI IWi'>C*

.

,,, i»^i,ilJn i t.-.pinv i_*|j^, ."Ci-b-.pijldu.,, Jt/itDi.j

lOr 1 <

m

i

K

Thijs (4 20) can Ic completed as follow;

!^'.>h

ri

«1. ' I'

QJIHiUi''!: ^

expi(m-i)hR

{]+(^i+iJui|/'i!Dll)}

1=1

for ! : i i m, and 1 i

m

< >1

1 evrin[(Kf,,-^:c^^^l'R'1*Pi H!^!/!:B!!}}

Proof

of

Lomms

4.3

'ror.'"; ''4 !9) 2nd Ismiris

42

vv'5 hsvs

Vm

^ !^-.t.b II'

J

'n'! 'W.^f' N,,,b II'

/

inllfW*

*i^rWlKj W'

<

(-fir. - ^:o)f^'5i.b''f'a-(l .n.+.b) li'^

yi

=

_C-iK^ f^tl!.. (A2r)

for

H

m

i

n.D

Ti'-i fnlln--^-iri'T rs-imr-lo+cc tho nrnnf of nuf_p \

M

2) fCT iTidSX OTlB 3'JStSITlS.

Lemma

4.4

Suppose _/^[rr.,33] li ncnsingular for 1 1

m

i >t: that is, J^^-*^ vi ncnsingular

for 1 i i 1

r

and k = mr. Suppose the hypothesis of

lemmas

4.1 , 4.2, and 4 3

are true.

Then

cur F. !.

—

-.11.- •

*

.1

1"

Is true for

aH

h e (O.Al.D

Proof

of

Lemma

4.4

i max{|i//„J! 1 _.

m

₁

K}

:=maviiiiii^T^ir,^^^^T]i|; 1

.mini

Now

let. us otjtdin _hixat- ^''-''''i ''^ 14) vv'e heve

!l>i^(fi)ll i he- !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 >:,. Ihui proving (421).n

Now

we

vv'1]1 obtain a bound for the expression defined in (4.19): that is,

Lennma

4.3

If t;'.8 hypothesis of len'in'ia 4.2 is true, then there e;iists a positive

constant C, such that

Recall trcrr; (4.17), and (4 23) that

<Cli(i'V<!! (431)

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 1 £ rn i H. Finally, recall

from

(4.1 1) that

lili"[nd,rr,]H

-

li>y[m.ddlikr,d,rn) '^[m,dalii[na,ml * l-^'^ _{Im,dd]l''^H^.Ind.mji!} < li/Vi^^,,,!! ilUr,, ,,3i! ^ _!l/y[m,dall! !!liina,m]!l * IH-^[m^dd]!"^ !! !l^Il^[nd,m]ii

= L,,{1 *

_L,,r.

_{L,,Z ,3(1 -L,,C}!|!il'}

_yi,

₍₄₃₃₎

for 1 i

m

1 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

_'W-'U,^Jy^

^U^CljjIKr)^.!!

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),

we

can proceed to

verify (43) - _{the first hypotriesis of lemrna 2.1} - _{as follovv;..}

Lemma

4.5

SupDCse

we

consider nornir.ear, decojp'.ed susterns of differential/

l^t^. dlL eqUJ.ibl.c. Ui .ilC ; Ui I.. .-. I ..-, 1Ui VVIIIwli li.I_3-' C^ji IC' -'I I-.. I u.!:, L:.

r.. .. t ".--,-•--he t-,.

.-.-*!•i.-'^ .-. 1c'Vir"-- =1-• zl 1 •''•'* '' 7 3>-i-' ' < «i>"i *r'ii

i.-.ri,..«,.,1. 0;-^^'-c= :.ilc ;!m-«,;.diii Oi .biiliiiai •4. 1 , "n.*., '<-'., Cilu '^.'^ C: c li Jc.

Then

the Frechet derivative, ir,Hi:\X ^^

_^/^^V-

satisfies (2.12), the 'irst

hypothesis ct lernrna 2.'. _ihai is, the mecivjaiity {A.-i'i is truc-u

Proof

of

Lemma

4.5

4.4.

Proof

of the

Second Hypothesis

of

Lemma

2.1

-LIpschJtz Continuity

of the

Frechet Derivative

To complete our ils^.tiity ar,cly5is, vve vviil ver;fy the second hypoiheiii

of eiMrr.G 2.1 that is,

Lemma

4.6

buppose the TuriCi.iCi". r_ in {i- ] }. is Lipschitz ccntiriiiou* vvit.."! reSpeCt _{tc £}

iiu _^j U.I i.-.(^,,..|-^pi. widi-ijwC-s LUC os-UIiiJ ^ui Liu; u^^i ival;vCi c-i ^;.r ;^:._ r_

With respe:t tc iiari:! z^ are ccntinucLiS.

Ther

(2.13; is true, that is, there e>;ists a constant Kl > 0, such that

VI .' . . " _ I .' ,M V' ' I !'" - II I'j "='• T.j; I'!•; k, - -rO''.i.'n••' and

Proof

of

Lemma

4.6

The vectorJlih satisfies the fclicwing deference schetTie

•"i-.r.. .'-' '«» ._i-..-. .f|- . --p- .\.t -i^D- -LI' r •'

i.;ic.^ri.2;' Sl.),iri<i .Tf:1 '-iri-3jlt.j,iri0,ri"ij ' iriOjUL.!,rri-1 '-"indllk. i.lrio.rn-l j' J- _r • . _

i'-' _{i-i.iriO.rrij} - _ii"

n.:.

r '• . .•• ... \ _ r J . ;

J_ij'-••r+l Jj-l.k-'l.. 2i_Vf+''' - —ill..-..•>"•-•

iLI.J UN ;;: |.^,. A

Wf

. . T ., , .T ,, , TIT

i.m .- lii.l,k+1 , JU.kfV'^ .. , _VVj^K+t' J' ,

kt . 1 ' .. T ... TIT

_j indx,-,] = liU,.3> + r.. .. , lij.d.k+r'j',

IsLJ. iriJ-rn] = ilVi.r+'i"'', , iv.i,kfr F,

Tj,[r,6/f,]

-

[rj,d>:+l''', ... ,rj^a,k+rT, J = '..-.. 1 < tTi : 'K, k = n-ir, and 1 : i ir.

Siiwtracting LUiterTi (4 35) from ;.Lii.tc;r; ;3 4) Uicl^s the fu!]ovv'ir'.9:

!hC'[r,,ji]"- [lii,[r,,j,rr.]-hQ(rQ]li',Jr,d,m]~P[r,^jjU, n-,-1 " f'it^inojli.uN.r.i-l]• = Ci.lrid.m).

£a^-V*T W-1-T .^.k-Tv ~ Ej-V-M' iiL)>+i' i^j>>i' = *^Ai.j..'i..i--i.

.

—

i- - -i

> Li ^-n I,iJ..

li,),iri-3.n'l ] = U.i,[I'lOxvj "

bi,t.if'O/n]

.

*'' ,iTfi/!"i ]

= wj^i ifi3,rri j ~ }v,i,[ri3,m]

,

li.1.[mJ.rn1 = ^1,[nd.mj "

W

j.(ri<i.r.-i1

iil^^n

-lii^y.

.^,.n'i'.

AT5^,,r.. = i^a^,,,,-'", ... , ij::^>.*r-'rr.

! . l.i i K = lili I .^ t . t -he i111c'.11 Ito.111w 1."^ -' >' _y ( CiIw^ iC' ,

'^•^indj'"' 'lii,M,r.-i]~f''-[aojil.i,[rid,m]"r'[nd]iil.n-i-1" -'''-indjliJ.fnd/fi-l ]' = L),[r.d/•-,),

f-r 'p. . -Vi.. Im. *''-r '-p ... 1,. , 4.r'.*r 'p , '\,"-- i..'

such t;-,G*

A t" . I.J.: .' 1 TC\

1 _ * .1 ~ '

Thui, thi Frechbt darivstive, L^Civ ,,h), or b^iiryf,) is defifit J e-such that K' r,-,-= L' '•-l.'-ojj _I' '*'L rid U r.-i,ridj , J/' -1 [flC(ndll _h^lf'^lj

1

V.rri L*rr. = [_hC{nc!]j P[ril Lirn ,-1 [hC[r.d]] [hF.lr.d)] Ur'l rr,^

jj.k+i -for 1 ••" I ^ r11 ^ I I , k = mr. and 1 i i i r.

',..,1.-. ...;

n

rr.'y ^.-.-.' z.i +1-..-, Hi

f

',^.-,k-r/-, .-' *^,- C^A-^^.i ,-i^. .-iii-i»-i•.- -. .-.f l" ,',,,

TTC TT I I I n UTT IUJr. U1. MIC UI I I CI C11LC UI 1.1IC I I CLM C'. U TI IVU'.1 VC O 'JI U.^.VJ^^ j-|

..'

(i=i,2) to obtein the Lipschitz condition in (4.35): that is,

where

r

f ,... \

./

( ... .

-

^1,rri " J^2/ri = 1^1 ,m

Wn^'^

' " i^2,m " ^'nn^'' J iJ_{rn Qrnd} .^''l .m"-it:,r."i '^ _1,m-^' 2;,rri r.-I .1 1

n\

-

p

. _{. .j'an'l-} .t+1 - i2.k+i 11 y*v - i2,k+-i

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 respect to v snc z<j are Lipschitz ccntinuous vvlth respect tc v and z<j

Ui; 1'

'11 .•''''fin'- 1''^-' l-'-wi c €..10^0 pUo;Llvc '«L>iic'.unlo r,^ V--I,~J, ^^u'! i.:iC>

and for

U

1 i r, k = Ku\\

m

. :M I !.,1 -_,j,_ ,' i -11'' ' 1 ^.'.'1 -...w .' -1 -IT"'. ...^ -^ -,^

il^h^i^Xh^ - U(i;s,h)!! i rr^a::!!:ir„(ji,^h) "-fm^Ji'2Vll^ 1 - --n - ^11

, _{-.-.:,...\-f.fli...}fi; -1 _{_} .-I If M A' _ ri II 1 , ,..., , .vf'l

- i--.t.f!u.-.;.Hi.

1 fT, ' 2,m''' 'I' 1 ,m '' 2,m''- '

-''-'

i--'

i Ifia;.{max{!;.Ji .^*^ - _{J^>^^IL IT}>*' - ]->*!! 1 1 1 ^ r, k = rnr,

u

m

. ni)

lrr.3^:!:•:.a..:;.,!!^, _,,, - ^v. ,,^!!,j..'W,^^,., - jr. „.,!:::iur,k=rn:-.,I<ru.ni!

< ir-ftv't i-\ ' II ... _ I... 11

: rr.a.>...r.,

, ^2. i.^i

i, Ji2.h''

_

I' II ,, _ ,,. I-

^2.hi--We

haVfe riov; tet. th'.ngi up to invoke Ifemma i ]

from

[kelTS]

To

arn.c gI

th;:. tO'.nt

we

sti'td arid i^ro'-.'t-l

vanoui

lernrra":.. These results are

summanzed

as follovvs

Theorem

4.1

Consider noriiinear, decouD'ed

systems

cf different!si/algebraic

eC'.iaf.ons of the fcrrn (3 1) Let

- 'mT - TIT

De a unique local analytical solution to (3. 1) Suppose the function

£in

(3 1}

;i c-^;. Li.;2.

i-La'' "-Xa' "'"-• '"-iLd- "-=<j» i^Ib ul lu.i I u..r> On

l.'.o,Afin'- ^^rr-^^^ <-»'C

nuDOthesis of lemnnas 4 1, 4 2, 4 3,

4

4, 4.5,. and 4 6 are true.

T-:.... . ,_ r rr-r -1--' ;>-.--T

.-;* -»•- * *-_r •-,-.-*I-- -«- i' < r- ^ -^ ^ ,- .^ *- ^-^»;..

—

—

I ,ici. v^.. r-i-rr OiO'^i. iifipiiLi>. Liifc-^cp Jit(.liw^ja '.. i.^.l Zt.'\..it'j \.L> n.^ni itlcdr, decoupled indei^ one systen'.s of differential/aljepraic eQuations of tne forn"; C3 1) are staple

G

5.

FSFF

Convergence

Analysis

for

Smooth

Index

One

Systems

of

Differential/Algebraic

Equations

Vv'e VYii: r.ovv utii-zt the results o&t5ir.eu in the previoui suDsectioni :o

prcve cunvtrcencc' cf our FSFF block implicit or.5-£twp

mcincui

(l 5)

applit-to smooth, nohlihear, decoupled inue;<: one

systems

of different'.al/slgep.'aic

equations of the form (3.1). Essentially, vve have set things ud to invoke

Theorem

2 2 from [4; to oDtcin the classical convergence result - _{vvhich v/ci}

stated

m

theorem 2 1 thot is.

Theorem

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) Suppose the hypothesis of

theoremiS 3.1 and ~- 1 are satisfied.

Then ior

some

pr, > 0, and

»-, •_ .'.• _ • ^•]/

the problem

has a unique solution v^, = v/^, e Sp;j(\y\X v«'hich converges to the analytical

LII-^'. I .'

r.-O ~

Ali-C, tl"ic crdtr of th'.s convergence is

th£l IS, there e::i2ts a positive ccr.st£r.t _{M„n. such} that Itvi. ih •"'h'' - ' _-con" •'-'

[1] 12] [3] 14] 15] 16]

References

Atkinson, K.E.,

An

iniruuuctwn tof\'umcncsJ Anslysis, John Wiley and

Sons,

New

York, (1978).

Frank, R , Schneid, J., and Ueberhuber, C, SiBtiiliiyProperties of

ImplicitRungB-k'utta n&thods, SI

AM

Journal of Nunnencal Analysis, (19S5), Vol.22, No.3, pp.

497-5

14.

Frank, R., Schneid, J., and Ueberhuber, C, Order Results forImpiicit

Rimge-Kuiia

nethous

Appiieu toStiffSystems, SI

AM

Journal of

Numerical Analysis, (1985), Vol.22, No.3, pp. 515-534.

Keller, H.B , Approximations Methods for Nonlinear Problems with

Applications to

Two-Point

Boundary Value Problems, Mathematics of

Computation, (1975),

Volume

29,

Number

130, pp.464-474.

\

I lauf,, I. n., L'Ji.'1/i i)ii(.>iiLii Ilemuus' lui ^uiviiiy i?!iiuiiiii onu

Discontinuous

Systems

ofDifferentisi/A/yeitrsicfgustions,. Doctoral

Thesis, (1986).

Petzold,L.R., Orderresultsfor implicitRunge-Kutta

Methods

Applied

toDifferentidl/AlgePrsicSystems,

SAND84-8838,

Sandia National

171

Watts, H.A., A-StatiJeBlock Implcit

Dne-Step

nethods.

SAND

J '

Date

Due

OCT.

08

^998

MIT LIBRARIES DUPl