ASYMPTOTIC ERROR EXPANSIONS FOR STIFF EQUATIONS. PART 2: THE MILDLY STIFF CASE

\?'.AUZINGER, R.FRANI( and F.MACSEII ")

Abstract. M7e derive an asymptotic expansion far the global errar of the imp!icit Euler scheme applied to stiff nonlinear systems of ordinary differential equations. Par t 2, in particular, is devoted to the mildly stiff case. I t is shown by a discrete singular perturbation analysis of the rernainder term equation that the desired order O(hq+') of the remainder term, wliich inevitably breaks down a t the first grid points, reappears a t t.he subsequent, grid points.

1. XNTRODUCTION

The purpose of this paper (Part 1 & the present Part 2) is to discuss rigorously the structure of the discretization error of the implicit Euler scheme applied to stiff nonlinear initial value problems. Our aim is to derive an asymptotic error expansion *")

with

(1.2) Ru = O(hq+') in the "B-sense" ,

i,e. the 0-constant is not allo~ved to be affected by the Lipschitz constant L of the right hand side f of the given differential equation but must only depend on the one-sided Lipschitz canstani rrt .

and on other prablem-dependent parameters of moderate size.

The functions ei(t) are solutions of the V.E.'s (variational equations)

and the reinaindcr ter111 Ru satisfies a nonlincar difference equation:

*) Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Wiedner Hauptstr. 6-10, A-1040 Wien, Austria.

'") For notatioiis not explained in Par t 2, as for instance h, C„ r ( t ) , . . . we refer to Partl 1 [I']. Formulas from Par t 1 will be quoted as [I,(...)].

(cf. 11 ,(2.10),(2.11) ,(2.12)]). In [I] we have sliown by a siiigular perturbatioii aiialysis o i tlie Ii.E.'s tha t the inhornogeneity b, - C , of (1.4j (ivliicli is defined iii terins of tlie e i ( l ) ) setisfies

(1.5) b, - C , = O(hq+') for E -+ 0 ,

where E > 0 is a small real parameter cliara.cterizing tlie stiffness (cf. [l,Section 41). (1.5) ineaiis that the full order q + 1 is present for suffi~ient~ly small E << h (strongly stiff case) a t all grid points. From this the main result of [I] , namely

(1.6) R, = O(hqt') for E -+ 0 ,

was easily concluded by est in~ates of B-type (cf. [1,(2.19)]). A structural analysis of tlie remainder term equation (1.4) was not necessary.

In the mildly stiff case E = h (more precisely: E not a t the O(hq+')-level), the Situation is different: Order redu.ctions of tlie quantities b , - C , appear a t the first grid points (cf . [ l ,~ect ion 31). The obvious coiisequence is an order reduction of Ru itself zt the first grid points. Numerical experience, however, shows that a t the 1a.ter grid points tlie full order q + 1 of Ru reappears by and by, i.e. the behaviour of Ru is cimilar to that of b , - C,. *) A proof of this phenomenon is iiot possible on the basis of B-type estinlates since

i) the bound in [1,(2.19)] involves max Ilbp - crll, p = l , ...,U

ii) in the non-sca1a.r case, rn depends on non-stiff eigenvalues of the Jacobian f „ i.e. m << 0 is not true in general; hence the factor Alerntv - 11 in [1,(2.19)] has no dampiiig power.

However, the dampingeffect mentioned above, namely Ru = O(hq+') a t tlie later grid points, can 'be prov'ed rigorously. Our proof, which is given in the present Paper, is based on an extension of singular perturbation techniques to tlie discrete equation (1.4). Tliis extension is by iio means straightforward. Recall (cf. [I]) that the functions e;(t) are unbounded near t = 0 : they are 0(c2- ' ) for i 2 3. This ilecessitated a singular perturbation ansatz involving negative powers of E. Such an ansatz is obviously not appropriate in the context of nonlinear eqiiations: If, for instance, the coefficients zi in an ansatz z = s W k x o + E - ~ + ' x l + . . . would be defined by equating coefficients of E - ~ , E - ~ + ' , . . ., and if x2 would appear in the nonlinear operator, then x2 = E - ~ ~ xo + ..., which shows that the ansatz fails. (The occurence of E - ~ ~ X ~ , for instance, would necessitate to equate coefficients a t - level, but then - instead of X = E - ~ X ~ + E - ~ + ' Z ~ + . . . - an ansatz X = ~ - ~ ~ x ~ + , . . woald be appropriate with x2 = E - ~ ~ X ; + . . . (!) and so on.) Thus - in contrast to the linear V.E.'s, for which an ansatz with negative powers of E could be applied successfully (cf. [1]) - such an ansatz must be avoided for the nonlinear difference equation (1.4).

For our discrete singular perturbation analysis we distinguish two cases:

i) The case E 5 Ch (where C denotes some constant of moderate size) will be discussed in Section 3. In Subsection 3.1 we will show that the inliomogeneity b , - C, of (1.4) can be expanded into powers of h (of Course, the leading term of this expansion is not O(hY+') as in the classical case but is a t the reduced 0(h2)-level). It would therefore be unnatural to work with (negative) powers of E in the singular perturbation ansatz for Ru. Our ansatz will be in (positive) powers of h, and the main result will be tha t 211 slowly varyiiig outer solution

*) The fact tha t for the quantities b , - C , the full order reappears with increasing U is a simple consequence of the fact tha t the solutions e i ( t ) of tlie 1 r . E . ' ~ become smooth with increasing t (when the inner solution terms aje dainped away.)

cornponeiits of R , a t C)(hi)-level, i < q + 1, l~ai~ish , such that onlp ~ t~ rong ly decaying iniier solution terms are present in the domain of reduced powers of It.

ii) The case h 5 CE (C a moderate constant) will ?ue coiisidered in Section 1. Iii tliis case the expansion of bu - C, in powers of h (as discussed in Subsection 3.1) brealis down. Avoidirig negative powers of E , are \vill do an analysis wliicli is based on s>izooth solutio~ts e,(t) of tlie V.E.'s, i.e. on solutions with moderate t-deriva.tives up to a certain order (and not affected with negative powers of E ) . We sliall theii obtain smooth inhomogenous terms bu - C , a t

O(h9;')-level. But , in contrast to the initial conditions e,(O) = 0 or e,(O) = ( i ) i~ ; (~ ) (cf. [I ,(2.15),(2.17)]), ~ h e starting values e, (0) are now fixed by the requireineiit tliat tlie e, ( t ) are smooth. These starting values influence the starting value Ro of the diffzrence equation (1.4): Instead of Ro = 0 ( or Ro = O(hq+')) we only have Ro = O(lt). Agaiii i t can be showii bg singular perturbation techniques (cf. Section 4) tha t the egect of the O(h)-startiilg value is damped away and tha t Ru converges towards a O(hq+')-solution of (1.4) with increasing V.

Summarizing the results of Section 3 ( case E < C h ) and Section 4 ( case h 5 CE ), we will observe t h a t our analysis Covers the whole "E - h - plane" in a satisfactory way.

2. THE TRANSFORMED REhaAINDER TERM EQUATION

Due to [1,(2.7s)], the iionlinear terrn j (~ , ) - R, in (1.4) can be expanded into

+ / f„ (L, .(tu) + vq (tu) + " . U ) (1 - o)do R Z ; 0

here we have used the abbreviation vq(t) := Izel (t) + . . . + hqe, ( t ) - cf. [1,(4.46)]). Using the denotation

1

G Y (Ru) G( tuI Ru) := /. fyy( tu> ~ ( t v ) + ovq (t,))do vq(tv) Ru+

+ j f „ ( tu, .+U) + vq (tu) + UR,) (1 - o)do R:. 0

and the representation [1,(4.49)] for the inhomogeneity bu - C„ we rewrite (1.4):

Due to our smoothness assumptions concerning f„ (cf. [ I , ( 4 . l c ) ] ) , G , ( R ) is Lipschitz continuous with a moderate Lipschitz bound; let LipG denote a cornmon Lipschitz bound for tjhe G , uniformly in Y . \Ve have

Now we transform (2.2) analogously as in [l,Section 41, using the transformation

( 2 . 4 ~ ) Tu := T ( t u ) ,

which diagonalizes f, ( t„ ~ ( t , ) ) (cf. [I ,(4.2a,b)]):

and premultiplying all terrns in (2.2) by T;', we obtain

1 1 T;'-(R, - Ru-1) = -[T;'R, - T , - I l ~ , - i + (T,-', -

( -2 .6~) h i z 1

= - ( P U - P u - 1 ) - @ h uPu-1 ,

We end up with the transformed rernainder term equation

(2.7) $ ( P U - Pu-1) = ~ ~ U P U + % P U - 1 + ru (pu) + 6" . Note the analogy between (2.7) and the transformed V.E.'s (cf. [I , (4.4)]:

(where A(t) = - T - ' ( t ) ~ ' ( t ) ) . In contrast to the V.E.'s, the difference equation (2.7) is nonlinear. Using the denotation

0, = 1 ( U r 9 l l 2 ( tu) ) ~ U ( P U ) = ( 71 (tu > Z u , YU) 9 2 , l ( t u ) 9 2 . 2 ( t u ) ) 7 2 ( tu > YU)

and introducing the "stretched stepsize"

we obtaiii from (2.7) after multiplying the second comporient by E = 5:

3. DISCRETE SINGULAR PERTURBATION ANALYSIS: TEE CASE E 5 Ch

In accordance with part 1 of this Paper, where the V.E.'s were considered in detail up to the fourth order, we will now (Section 3) restrict our considerations to the case q = 4 (i.e., the remainder term R u is of the order q + 1 = 5). The extension to a higher order is straightforward but techniqually nontrivial.

1- 3.1 h .- Expansion of the Inliomogeneity 6,

First of all, the inhoinogeneity 6, of the traiisforrned reniaiiider term equation (2.7) is expaiided intopowers of h on the basis of the singular perturbatioii zxpaiision of tlie ei(t). Due to [1,(4.49)], (2.4) and (2.5), 6, reads for q = 4:

E where the nonlinear operator I F U ( v ) is defined by

6, splits into smooth terms (depending on t u ) and stretched-variable terms (depending on r, = $): With

(where &(t) and Si(;) denote the outer and inner solution cornponents of the E; (t) (ci. [I ,(4.38)])), we have

(3.3a) 6, ='6ym + 6Etr ,

where 6;" is the smooth part ")

and 6$r := 6, - 6,Pm reads

5 + I F , ( ~ ( t , ) ) [(T, ~ ( t , ) -i- T , ~ ( T , ) ) ~ - (T, ~ ( t , ) ) ' ] .

*) A,E(~,) is asinooth term (despite A u = O(E- l ) ) as has already been shown in Part 1 (cf. j1,(4.50)]).

The bilinear operator B ( t u ) is the sanie as in [1,(4.69)]. Note tliat

as has already been shown iii (1, Subsection 4.3). 011 the contrary, 6Eti shows order reductions a t the first grid points.

We shall now expand 6ytr into powers of h, proceeding from the following h - expaiision of tlie inner solution part S ( r ) of 6, ( t ) (cf. (3 .2)):

4 4

4 C h ' ~ ~ - ' m o , ~ ( r ) + C h%4-'rnll;(r) + . . . ( ) ( ) 4 i=l 4

i= 1 C h ' r 2 ~ ' n o , ; ( r ) + C l ~ % ~ - ' n ~ , ( r ) + . . . (3.5)

i=l i=l 4 4

h S C ~ ' - ~ r n ~ ~ ; ( r ) + h4 C ~ ' - ~ r n ~ , ~ ( r ) + . . .

=

i r -1 i=l 4 4

h2 C ~ ~ - ~ n o , i ( r ) + h3 C ~ ' - ~ n ~ , ~ ( r ) + . . . i=l i= 1

Here we have used the E - expansion of the ei( t ) (cf. [1,(4.29a)]); recall that W = (cf. (2.9)). VS7ith the abbreviations

(3.6n) M L ( T ) := E w ' - ~ - ' m i j i ( r ) , N L ( r ) := E wk2--I n i , a ( ~ ) , 1 = 0 , 1 , 2 , . - - , i=l i=l

(3.5) reads

(3.6b) ,!?(T) = h3 MO ( T ) + h4 Ml ( T ) + . . . h 2 N o ( r ) + h3 h~~ ( T ) + . . .

\7i7e shall need the usual expansions around t = 0: *)

1 1 @, = @ ( t u ) = -- -T- ' ( t , )[T( t , ) - T ( t u - h ) ] = -T- '( t ,)T1(t ,) + h T ~ - ' ( t , ) ~ " ( t U ) + . . . =

h 2

(see [1,(4.64)] for the denotation A( t ) , A ( t ) ) ,

( 3 . 7 ~ ) B ( t U ) = B ( O ) + h v B 1 ( O ) + . . . ,

(3.7d) = h ( X ~ l l ( 0 ) + h ~ X ~ l l ( O ) + . . . + hw-l[Xl , l (o)+huX~ll(-o)+" ' ]+ hu - l [Ylli (0) + huYi l i ( 0 ) + . . .] + . . .

hXal i (0 ) + h2 [uXhl l (0 ) + w-'Xl,l(O)I + . . . ) + o ( h 2 ) h2[w-lY1,l ( O ) ] + . . .

") I t will turn out that the terms of these expansions always appear in conjunction with expoiientially decaying inner solution components; since terrns of the form "U' . exyonentially decaying term" are bounded uniformly in v (a t O(1)-level), terms like " h i . v i v ezponentially decaying ternz" are indeed a t ~ ( h ' ) - l e v e l . The Situation is analogous as in Part 1 where "t" inner solution term" = "E' T ' -

zmer solzltion term" was considered o(E').

(Cf. [1,(4.29a)] for the definition of the Xi j ( t ) , &,; ( t ) . )

(3.6b) and the expansioiis (3.5) are now inserted iiito ( 3 . 3 ~ ) ) and tlie secoiid coinponeiit is multiplied by E = 7iw-'. Rearranging in powers of 12 we obtair. the h - expaiisiori for 6Ytr :

where the leading terins read

and

For the terin ori inating froin B(t,) we have used the abbreviation 6 (3.,8d) b(Q) := b i 1 2 (O)Xo,i(O)

in accordance with [1,(4.70)]. For the present purpose '(disc&sion of remainder term R, of fifth

order) we shall need the explicit representations of the leading terms l$;l(f; (cf. (3.8b)c)) bu t not

g f t h e ~ g ! , j > 2.

I t should be pointed out that the expansion (3.8a) is not to be understood as a pure asymptotic

expansion in powers of h; tlie coeficients 1:: are thernselves h-dependent (via the para~neter w = S ) . Powers of w appear in the l(3j explicitly (cf. (3.8b,c)) as well as implicitly within tlie ML(T) and Nl(r ) (cf. (3.6a)). Thus it is not a priori obvious that our arrangement in powers of h (cf. (3.8a)). is really justified. (For instance, positive powers of w become unbounded with increasing stiffness, i.e. with E + 0.) Let us discuss this point iii rnore detail:

i) Due to assumption E 5 C h , any negative power of w is harmless because w - ~ < ck. Tlie

positive powers of w which appear in the 1Y) sre harrnless, too, for the following reasons: Each k term W ' ~ ~ , ~ ( T , ) or w nl, i(ru) (where ml,; (T,), nrli(r,) are inner solution terms of the form

p ( r , ) e - c ~ ( o ) T ~ ~ ) is uniformly bounded for E + O a t all grid points v 2 1 due to the exponential \ damping ( cf. [ ( ) ) However, a problem arises for u = 1 because ML ( T , , - ~ ) = Ml (0))

NL (ru- 1) = AT1 (0) appear ( for which e-"2(O)'~ = e0 = 1 has no damping power ) . But it can easily be seen that those quantities rn1,;(0), nlli(0) which are affected with positive powers of w vanish due to the " ~ ~ - p r o p e r t y " which has been proved in Par t 1 of this paper ( cf. [l,Subsections 4.3, 4.41). Hence (3.8) is a reasonable expansion for E < C h .

ii) For h < C r (C some moderate constant) the situation is contrary. In particular, negative powers of w blow up for h << E , the consequence of which is that only

cali immediately be concluded fronl the above representation. This shows that our represeil- tation (3.1),(3.3);(3.8) for 6, is not suitable for the subdomain h 5 CE of the " E - h - plane". (In tlie classical case - E not sma11 - for instänce, it is well kiiown tliat 6 i m as well as 6 i t r are a t 0(h5)-level.) In Section 4 the case h 5 CE will be treated by a different approacli.

~ F o r 6ym = 0 ( h 5 ) we will use the expansion

3.2 Ansatz for p , aizd Equatiiig Coefficients in the Remainder Term Equation

For p , = (i.) a7e make a discrete Singular perturbation ansatz ic poxers of h:

X:) , yJi) are discrete analoga of the srnooth outer solution terrns, and C;) ) will be understood ( R ) as discrete, rapidly decaying inner solution terms. For the remainder terrns X, , of the

expansion (3.11) one would expect that estirnates = O(h6), yLR) = O(hG) can be derived.

-4s the discussion below will show (cf. Subsection 3.4), only xY) = O(hS) , = O(h5) holds, w'hich is of Course sufficient for our purpose (recall that our aim is t.0 sliow that p, = 0 ( h 5 ) with increasing U). (This situation - tha t one power of h "is lost" within tlie (straightforward B-type-)

( R ) y y ) - is quite analogous toePar t 1 n~here one power of E W ~ . S lost withili tlie est ihation of X, , estimation of the remainder terms of the E - expansions of the ~ , ( t ) ; cf. [1,(4.24)].)

'Up to nbw we have not fixed the starting values xo arid yo. RecaH tha t our difference equation is considered in an interval [O,T] which is interpreted as a subinterval (with constant stepsize h) of the whole integration interval (cf. [1,Section 21). The starting value for the remainder term Ru is inductively defined by the remainder term on the preceding interval (cf. [ , ( 2 1 7 ) ) Only for the first of these subintervals the starting value is 0. Anticipating the main result of the present paper - tha t a t the end of each subinterval all order reduction effects are damped away - we inductively

i.e., the starting conditions for the X:), Y';), <('), yLR) of our aiisatz (3.11) read

xAo) = 0, yO0) + = 0 . . . 1z2- level

X;" + = 0, yil) + = 0 . . . h3- level - ( 3 , 1 3 ~ )

xO2) + E; ' ) = 0, y o ( g ) + ,1;2) = 0 . . . h4- level 1

(Xi3) + <i2)) + zLR) = 20, (yO3) + + yiR) = ,Yo . . . h5- level

' ~ i r i ' c e r f i i n ~ the freedom a t the O(h5)-level (last line of (3.13a)) i t will turn out to be appropriate (for. techniqual reasons) to choose *) ., : :

7 . .

*) 'As 'mentioned above, it will turn out that sLR) and I J ~ ) are only O ( h 5 ) but not O(hG).

t.beherefore natural to choose x r ) and yOR) a t O(h5)-level.

Now W& proceed as usual in t8he singulai perturbatioii t,lieory: The aiisatz (3.11) aiid the expansion for tlie inhornogeneity '6, (cf. (3.3a)) (3.8a) and (3.10)) will be inserted into tlie difference equatioii '

(2.10). Our main goal is to show that all order reduction effects disappear with increasing U , wliich

means that the quantities X?) and Y;') must vanish for j < 2, sucli tha t smooth outer solution coqponents a.ppear a.t 0(h5)-level only. Our proof of this damping property is based on algebraic investigations of tlie Ieading iiiner solutioi~ terms. The "kernel" of our argun~ent~ation will be tha t the usual requirement lim {Y) = 0 must be compatible with a starting condition (i3) = O for

,403

j = 0,1 . From (io) = 0 and (Ai) = 0 and (3.13) tlie conclusion X;') = 0, X;') = 0 and xi2) = 0 can be drawn; then it will be easy to derive tlie desired property X:) E Y(') = O for j 5 2.

Sher i are'several technical points to be discussed. Let us consider in detail the nonliiiear terrns ri[tY , > , i ~ Y , Y,,) of (2.10). With the denotations

X, := h 2 x Y ) + h3xL1) + h4xL2) + h5X(3) , (3.. 14a) I ,

1% := h2yJ0) + h3yj1) + h 4 y j 2 ) + h5yL3) ,

and

F

these nonlinear terms read

(3.15) ri (tv, P,) = 7i(tv, &v + P?)) =

= [ri(tY,&Y + P?)) - r i( tv)&u)] + [7i(tY,&cm + & Y r ) - ri(tY,PEm)]+ ri(tU)PYm) .

According to the first bracket on the right hand side of (3.15) we defiiie the nonlinear function

V,;, (P) 71(tu:&v i- P) - 71(tu,&u) ( 3 . 1 ) I$(P) = (V,;2(p)) := .;L + P) - 72(t.:Pu) ) *

The properties of V,(p) will be essential for the estirnation of the remainder term p y ) = (;F;) '

I

qf the ansatz(3.11). (Wlien pLR) is to De discussed, &, is already deterrnined due to the recursive structure of tlie singular perturbation technique, i.e. V,(p) - depending on &, - is then a weI1-defined function.) Due t o the definition of the r i ( tY , X ) Y ) (cf. ( 2 . 6 ~ ) and (2.8)) and due to our smoothness assurnptions with respect to f (cf. [1,(4.2c)]), the functions V,(p) are Lipschitz continuous for all Q: Let Lv denotre a Lipschitz bound which holds uniforrnly in V:

. "

(sb16b) IIV~(P) - Vu(~)ll 5 LvIIP - ~ 1 1 -

In p~r t i cu la r , due to the definition of V, (p) (cf. (3.16a)):

(3.164 L ( 0 ) = 0 , IlV,(P)ll 5 LvllPll .

Our next Step is to expand the last term rt(tv,PYm) on the right hand side of (3.15):

10

= Y; (t,, 0, 0) + 7i,(t,, 0, 0) ( h 2 ~ ' ' ) + . . . h . 5 ~ y ) ) + r„ (t,, 0 , 0) (h21'J0) + . . . ~ z ~ y ( ~ ) ) + 1 2

( 3 . 1 7 ~ ) + h4 Sr ixx( tv , 0, 0) (XI?)) + h4yixy(fv, 0, O)XY)-I '(O)+

+ h 5 ~ i x y ( t y ) ~ > ~ ) ( ~ ~ ) ~ ( ' ) + -Y(')Y(O)) + h5Tiyy(ty > ~ > o ) - I ' > O ) l i ( ~ ) + 0 ( h G )

Froin (2.3) and ( 2 . 6 ~ ) we have imrnediately *)

(3.17e) All higher derivatives are 0(1), too.

From (3.17) we can conclude that no difficulties arise from the iionlinear terms r i ( t „ er): Tlie

quantities X?) and Y")) which are defined by equating coefficients of h j+2 , appear a t least a t ~ ( h j * ~ ) - l e v e l in the expansion (3.17a) (due to (3 .17~) ) . Hence, due to the recursive nature of the

singular perturbation technique, these terms do not influence the definition of the X?) and Y(') themselves; they are only inhomoge-neaus terms in the respective difiere~ice equations defiiling the . X Y 1 > . (h?any of these terms will actualiy vanish since we shall sliow X?) G Y(') I 0 for j _< 2.) One point which is of minor importance for our purpose has to be mentioned briefly: The terms ~ ; , ( t „ O , O ) , r „ x ( t , ~ ~ , ~ ) , . . . which occur in the expansion (3.17a) in conjunction with

( 3 ) (j) non-vanishing quantities X, , Y, have to be split into smooth and stretched variable terms in the usual way - for instance yix(t,, 0,O) = r:r(t,, 0,O) + r Z r ( t Y ) 0) 0) - according to the dependence of the ri's on ü4 (t,) = E(t,) + s($) (cf. for instance the splitting of 5, in (3.3)). Terms originating

from the smooth components of tliis splitting - as for iiistance f F ( t , , O , O ) X ( ~ ) ) j > 2 - will appear (1) ( 1 ) in the inhomogeneities of the differeilce equations which define the outer solution terms X, ,Y, ,

and terms originating from the rapidly decaying components - as for instance 7 $ r ( t „ ~ , ~ ) ~ Y ) - are involved in the inhomogeneities of the difference equatioils defining the inner solution terms

( 1 ) (1) J, . Surnmarizing, we observe that none of the terms of the expansion (3.17a) has an influence on

the definitioii of the ~ ( j ) , Y(') and the C?), thernselves - they only appear as recursively

1 *) With the denotation W E (:::: := T;' f y y ( t y ) i ( t y ) + a T u ~ 4 ( t v ) ) d a T , v ~ ( ~ , ) - T, for

w2,2 0

the rnatrix appearing in the first term of ( 2 . 6 ~ ) ) a e Iiave &(W .p) = &(W ( y ) ) = and W 2 , l

*(W a~ = L ( ~ . l i . 3~ ( i) ) = (t:::) ; note tha t MT = O(h) due to the factor ur (t,) = O ( h ) The

~ a r t i a l derivakives of the second term of ( 2 . 6 ~ ) (with the bilinear o ~ e r a t o r ) obviously vanish for p = 0. Thus (3 .17~) holds. (3.17d,e) is an immediate consequence of our smoothness assumptions

. \r,it .f (cf. [1,(4.2c)]). L i b - l .

- .

11

defined inholnogerlous terms in the resp. difference equations. Recall that our aim is tlie explicit deterrniiia.tion - with algebiaic rnanipulations - of the leading iiiiler solutioii ternis and to sho\v

~ ( 3 ' ) - that the starting condition (F) = 0, j = 0, 1, is compatible witli lim 6 , - 0. To tliis eiid u+03

we shall explicitly determine the stretched- variable coefficients a t the h.2- and h3-level. Siiice the second equatioii in (2.10) is multiplied by E = hw-', there exists only oiie stretched-variable teriii

of the expansion (3.17a) a t O(h3)-level, namely riz(t,,, 0,O) h 2X(0) - - O(h) . h2x(0). I-Iowever,

when equating the coefficients of li2 we shall show that X?) 0, hence r l , ( t „ ~ , ~ ) h 3 ~ ( o ) will vanish aiid there do not exist tcrms of the expansion (3.17a) which influence the leading terms (j) (3 )

[V A U , j = O , l .

As a final remark concerning tlhe expaiision (3.17a) we note that a simple stability coiisideration

will show that the non-vanishing terins 16') (afhich depend on h in a harmless way) are 0 (1) uniformly in h. This will serve as a justifica.tion for the fact that we have written O(h" for the remainder term of the expansion (3.17a).

Let us now continue the discussion of the splitting (3.15). The second bracket on the riglit hand side of (3.15)) which is of purely stretched-variable type, is treated in a similar aray as -y;(t„ ;Y):

For z b expansion (3.18) essentially the Same remarks apply as for (3.17a): None of the terms in (3.18)

dcie's influence tlie definition of the (Y), q()) themselves; they only appear as recursively defined i>hornogeneouc terms in the difference equations defining the E!), q f ) ; I > j . All terms in (3.18) are of Course assumed to be expanded around X = 0, y = 0, a.s for instance

(The leading terms h 2 ? y (t„ 0, 0 ) q p ) and h2-y$'(t„ 0, 0)qY) are actuaily ~ ( h " due to ( 3 . 1 7 ~ ) ;

all otlier nonvanishing terms are at least at ~ ( h . ~ ) - l e v e l since X; ) , Y'') , O for j < 2 which will be shown below.)

There renains one point w.r.t. the expansions ( 3 . 1 5 ~ ~ ) and (3.19) to be mentioiied: All smooth coefficient functions appearing in the equations for the inner solution terms are to be expanded around t = O (cf. for instance (3.7) or [1,(4.11)]). Hence, for instance, -yfr(t,, 0,O) is to be expanded into *)

* ) The expansion (3.19) contains, among others, a non-vanishing term h ' h 5 y i w ( t w 0 , ~ ) ~ ' 3 ) ~ ( o ) j afid

1

It is obvious tl-iat only snzooth data functions are to be expanded around t = 0. It would not be reasonable to expand stretched - exponentially decaying - t e r~ns (as ior instaiice r$'(t„ 0,O) or

B: (0) , ) around t = 0 due to their lack of smoothness.

Recall tha t , for our purpose, the ezylicit forin of the stretched-variable terms in the espansioils (3.17a) and (3.19) is onlg needed a t O(h2)- and O(h3)-level (siiice u7e will explicitlg determine

1. (P), V(') and C('), - cf. Subsection 3.3 below). Taliing into account that the second equation in (2.10) is rnultiplied by E = hw-l, we see that there is only olle such term a t O(k7-level, originating

from h271, (0, O;O) r lY) iii (3.19). (Note that the 0 (h2)-level is not influenced by the nonlinearity.) Using the definition ( 2 . 6 ~ ) of J?, (p,) and taliing into account *) tha t i4 (0) = hXol1(O) + O ( h 2 ) , a

simple calculation shows that the only term a t h3-level originating froin ~ z ~ ~ ~ , (0, 0 , 0 ) ~ ~ p ) reads

(1) (for the definition of b(0) = ~ : , , ( O ) X ~ ~ ~ (0) see (3.8d)). (3.21) will appear in the equation for E, .

I For the collection of all other terms of the above expansions (cf. (3.17) - (3.20j) a t ~ ( h j ) - level, whicli do not influence our algebraic manipulal;ions, we simply write

(3.22) hj( I ( j - 3 ) ~ m + I(3;-3)str) . L';% 3 ,

here-the subscript i (i = 1,2) denotes tlie first or second cornpoi~ent, respectively. Hence the yi(t„ P,) from (3.15) have an expansion (for the definition of VVii(p) cf. (3.16))

4 (1)str ~ l ( t , , p , ) = h 3 ~ ( ~ ~ s m + h 3 b ( ~ ) r i ( 0 ) + ~ 4 ~ ( ~ ~ s * + h I,;l + ( 3 . 2 3 ~ )

5 (2)sm 5 (2)str

C + h I,;, + h I,;, + vV;1 (ptR)) ,

3 ( o > ~ m + h31(ojstr + h41(jjsm + h41(iastr+ 72(t„ P") = h I,,, (3.23 6 )

+ h5~(;z)"* + h5~(Sz)Str + V Y ; 2 ( pLR))

S h e terrns j = 0 ,1 ,2 , will turn out t o be Zero since X:) Y(') 0 for j = 0,1 ,2 .

With these preliminaries, the transformed remainder term equation (2.10) reads:

h 2 A [ ~ ~ ) - X;,] + . . . + h 2 [ ~ ( 0 ) - ~o , ] + . . . = h

= c l ( t , ) + [h2x(O) + . . .] + [cl(0) + hucl(0) + . . .] . [h3((0) + . . .I + + O1,1(6,) [ h 2 ~ ( 0 ) 1 + . . .] + [a l l l (0) l h(va:ll(0) + ä1,1(0)) + . . .] [ h 3 c O 1 + . . .] +

2 (0) (3 .24a) + 91,2(tv) . [ h 2 ~ ( ? , + . . .] + [al,2 (0) + h ( ~ a : ~ , ( 0 ) + ai lz(0)) + . . .] [h 17,- , + . . .] + 2 (0) ? ' (0)sm 3 (1) q (1)str + I,;1 + h'IVil + h3b(0)17(0) + h l,;,-+ h 4 ~ ~ f ~ s m + h I,;, + 4 ( 2 ) 5 (2)sm 5 (2)str + h I,;, + h I,;, + hkI,;,

5 (7) + h5k(oi + h + ~ ~ ; l ( ~ y ) ) + + terms whieh are at least at 0 (hG) - Level ,

xi3' -X,(,=' therefore it seerns t o be necessary to expand xL3) around t = 0: xL3) = ~ f ) + h v + . . .; such a "discrete Taylor expansioil"' does indeea exist with a remainder term which is uniformly bounded a t a certain hR-level. Nevertheless, such an expansioii is not required here because all respective (non-vanishing) terms are a t least a t 0 ( h 6 ) (remainder terrn-) level. (Within tlie equation

for the remainder term no expansions around t = 0 will be used.) ") Cf. (3.7d) and recall tha t the inne; solution terms of ü4(t) appear with higher powers of h: For

instance, h r 2 m o l l (b) = ha~-2,0,1(%).

and

1 ( 0 ) G 1 ( 4 ) h + h6-ku,2 + h -Iui2 + --vu;2(pLRR))+ W W W C

r- 3

+ terms which are at least at 0(h7)- level . I ) - - 3 .

r L

hq&tirig coefficients in (3.24) and using (3.13) for the definition of the starting values we obtain:

Thus, Y'O) 0. From (3.13) we have X?) = 0, which yields X'') 0.

[p - ((0) ] = (0) (0) U - 1 a i , 2 ( 0 ) t l v - i + L;,,

(3.256) 1 * M

(Note the interrelation between the coefficients in- (3.25b) and (3.86).) Frorn Y:') 0 and (3.13) we have = 0. will be explicitly determined in Subsection 3.3. As usual in the singular

perturbation theory we shall fix the starting value cp) such that lirn ero) = 0. It will turn out in U*OO

3 & 6 s ~ t i o ~ ~ 8 . ' $ ; that er) = 0. Due fo (3.13) this entails = 0. f,p"-*-.. ?+:"

' - I t i %

: - . , 3 . . , > ' ' ' I

1 (0) E 0, ;8211(tv)X,-1 E 0, B$~&:XJ') -5 Y$'' = 0 and (3.17a-C) we irnmediately obtain I,;,

(0) @~:2;j.(t,)~,-l 5 0, ~ ~ [ Y J O ) - Y'!)~] EE 0. Therefore:

Again, I$') 0. From Xi1) = 0 we conclude X(') 0.

. f

[E<" - E"' ] = (1) U-1. a'i2(o)>7"-l+

, . - .., _. . + c ~ ( o ) E Y ) + ~ ~ , I ( O ) E ? ~ + [valt2(0) + ä1,2(0)1qz1+

(WJote the interrelation between the coefficients in (3.26b) and ( 3 . 8 ~ ) .) From Y:') = 0 and (3.13)

we have V;') = 0. Again we shall see in Subsection 3.3 that the condition lirn E(') = 0 yields U 4 0 0

5:') = 0. Thus, xi2) = 0.

" . Coe ficients of h4:

(1) From X(') E Y(') n E E 0 and (3.17a-C) immediately l(;jsrn 0, t t 9 2 , 1 ( t v ) ~ v - = 0, (1) $t92,2[tv)~v-1 0, 5 - 0. Therefore:

1 ~-[x(Z) - = c l ( t , ) ~ ( ~ ) + d l , l ( t v ) ~ ( ? l + g1,2(tU)y'!)1 i

C3.27~) .. . .-*? - - L 7 , -. , .-- ?..

. . 0 = -c2..ftv)Y, ( 2 ) ,

U.?- 4 -, ,+",?!. . > s-

Again, - )C' 0. From xA2) ='o we conclude Xi2) E 0. . .

J. *

. [E:) - = a i , z ( ~ ) ~ ( z ? ~ + PO(V)S?) + ~ i ( v ) t i O ) + (0) + p i ( v ) t F l + PO(V)E'-~ + P I ( v ) v ( ~ ~ + P ~ ( v ) v ~ - ~ +

+ 1(?lstr + 1:; ,

p;(v) is-usea as a generic denotation for polynomials 'of degree _< i in V . E.g., (V) within the

tepm pl(v)@) (first component of. (3.27b)) denotes the quantity vc:(O); p1(v) within the term 1, ; p i ( v ) ~ p ? ~ (secöiid compohent of (.3.27b)) denotes the quantity (v&:,~ (0) + ä2 (0)).

C? -

The starting value for r,~:~) is rlh2) = 0 (cf. (3.13)) because yi2) = 0; is again determined by the

ecpdj&i~q ,J..p& = 0-. In Subsection 3.4 we shall see that tL2) and are uniformly bounded

and d~qay? d l like P ~ ~ v ) ( I + wcz(0))-".

h i i i f t ~ ~ r ~ b r 6 the starting value for X;) is determined by Xi3) = 0 - eh2) # 0 (cf. (3.13)).

~ e ~ i e q ~ + t ~ o f i ~ : ; i ~ - f i ' f i i n g xL3) and x (~) read (again we have I ~ ~ ; ! ~ ~ E 0 due to X;' G Y(') , 0 for jV= o,;&,'i);:. *<S,{ :q::;, i

(3) (3) $ 1 r [ ~ p ) - xO1] = c l ( t , ) xy ) + di , i ( tv )xu- l + d l , ~ ( t v ) Y ~ - ~ + k c i , (&2:28a)" h 0 = -Cz(t,)~'3).

(3) '2 = $0 , D ue to k:! + 0 and Xi3) = 0 - [g2) # 0, X?) does not vanish. But the

boundedness of xL3) (uniforrnly in h) will become clear by a simple stability argument; it is essential

l%k~-if) *. I is bounaed. (Cf. Subsection 3.4 for details.) 2 .

11 C

For theinner solution a t h5-level there is only an equation defining ( E?) is already defined . L ...

b i t l 3 . bb ) , and a quantity [Y) at hG-level does not appear in our ansatz (3.11)): - ' < . $4 : . , , n

(3) T&e&+gtipg yalue for (3.28b) is rlr) = 0 - yi3) = 0. In Subsection 3.4 we shall again See tha t V, _ , - . "

1q;uqif~mly bounded and decays like pi (U) (1 + wc2 (0)) - Y . 1.b .' * . - %

.-- ,

Remainder term P!?) of ansatz (3.11): : r

%'*~%tain the desired estirnate p y ) = 0 (h5 ) , the difference equation for is re-scaled t o its original f&; in particular, the factor E = by which the second component is multiplied (d. (2:1(3)) - X ?L is now omitted. This results in a differente equation of essentially the Same type as (2.7):

. .

(3.29) $lpy) - p;R)t] = bUPY) + + vv(p?)) + 6LR). " "

i - 'rhe inhomogeneity 6jR) consists of all terms which do not depend on p y ) and have not yet appeared in the equations (3.25), (3.26), (3.27) and (3.28). In Subsection 3.4 we shall show

.. 3 -

(3.30) pik) = 0 ( h 5 ) ;

b,g~a &a$iärd B-oonvergence estimate. To prove (3.31) it will be essential that nonnegative powers of w = 5 appear only together with exponentially decaying inner solution terms in 6LR). But note

$kpk,6LR"-' $+.W O(hB) is not true because the second component is re-scaled by the factor X . (For

instance,*~%','cbhtains the terrn h5<92,i (t f 0 which originates from h i d2,i (t,) h 5 x F l in (3.24b) .)

"=4 "; t ; ' L "

@t' &*&um&&&k: .Once it is proved that lim [P) = 0, j = 0,1, is compatible with cg) = 0, V-+Oo

j z - s F O , I,! i ~ e . sl&ly vary ing terms X;) , Y;') of the ansatz (3.11) turn out to vanish for j < 2

(rnoreover, yL3) E 0). Hence (3.11) reduces to j .,% , .- ' ,

(3.11') PP,,= h5x '3 ) + h3tL0) + h4<y) + h5[L2) + zY)

_ s. h2qLo) + haI)(l) + h4rlL2) + h5rl'3) +

F r n ~ ~ g ~ h i ~ the desired structure of p, immediately follows if the non-vanishing inner solution terms

(F), +,- . k ,

are proved to be of the exponentially decaying form pi(v)(l + W C ~ ( O ) - ~ , if the non-

v&8&hing outer Solution term X?) is shown to be bounded uniformly in h, and if (3.30) is verified.

411 C- these , points ,are discussed in detail in Subsections 3.3 and 3.4 below.

3.3 Explicit Determiiiation of the Leading Inner Solution Terms , >

I We shall now give explicit representations for the solutions cp ) ,qp ) and E?), q?) of the equations (3.25b) and (3.26b). In particular, we will show that = {Ai) = 0 i s compatible with lim {Lo) =

V+OO

C iim.' - &. As we have seen in Subsection 3.2, this " darnping property" , which isr in a certain ,P4= . -

% : , )., , - t %&;P sense dhafogoiis to the "E'-property" of the e i ( t ) (cf. [l,Section 41)) is the crucial point in our qnalysis.

: T

b 2

In the sequel we will use the abbreviation :' r .

(3.32) Q := 1 + wh2(0) .

Note that IQI > 1 due to our assumption Re(c i ( t ) ) > 0 (cf. [I, Section 41).

, . 1 - Deteiinination of E!,'), nY):

With (3.8b), the equations (3.25b) read

(0) *"C" -fb"' * col = U ~ , ~ ( O ) ~ , _ ~ - [MO(ry) - Mo(r,-l)] + ~ l , ~ ( Q ) N o ( r v - ~ ) 9 .

(5.33) 1 =[qp) ' (0) 1 > < + - ui V,-,] = -&(o)tlP"- -[No(%) - No(~v-l)l - CZ(Q)NO(TV)I . 3 s - J < < W

With

;sLj.,'. ,

f3.331 is equivalent tb

L.'-<"-l8. >#,O) - p1 = alt2(o)ij(: X$..SJ V 1 ,

!Li>? \ ~q;?) - fl;21 = -uc2(o)q;02) ,

e~g($;t6 = 0, we obtain (cf. (3.34)):

By linear combination of the equations in (3.35) (where the second equation is rnultiplied by - ) and with the definition I? i.4 'a

(3.37) ii0) := ( L O ) + wc2 (0

we obtain

The gengral solufion of (3.38) is

(339) $O)' = -a1,2(~)i7L0) + C .

Thus, due to (3.34)) (3.36) and (3.37))

Sime tim Q-' = lim Mo(rv) = 0, t e condition lim = 0 requires C = 0. Hence tLo) is V-+OO '-+W '-+W

fixe& the starting value is

(3.414 [F' = -Mo@) - +wc2(O))No(O)

Vciiiii (3.6a) this results in

4 (3.42) $ h i A 3 h i , 2 ( ~ ) n ~ , ~ - l ( ~ ) = ~ ~ ' - ~ a l , 2 ( 0 ) n o ~ , - ~ ( 0 ) ,

i t 2 i=l J x" X, :;J+:;

b$&ab~e +LO 6(6r ="O by definition (in accordance with [I ,(4.72)]) and no14 (0) = 0 which holds due t&~%hc?@'~>$cfpert$ .(cf. [I ,(4.93b)]). Hence we obtain

r :i. &-T$ L *>- i. . ,

4

= - C wi-3 [mo,;(0) + (o) (%;43J+t ,Ep i=l al."o) (no , i (~) + c2 (o)no,i- 1 (o))] = 0 ,

because ?o,,{O) + (no,, (0) + c2 (0)noli-i(O)) = 0 for all i 2 1 as has been shown in Part 1 of tbs;eqp.per (cf. [I ,(4.78)] and [l ,(P.73b)]).

Determination of E( ' ) , V" ) : 4 8 g @3~ f%8c j j 'the equatidns (3.26b) read

$ 2 51:

Witb (3.34) and

{&$gJi: . + cl (0)EP) + a i l l ( o ) < ~ l + [vaiI2(o) + ä1,2(0)]~f!~ + b(0)iiY) , (0) t%rtkirr .i$$l) - tf(121 = -WC2(0)qp) - wvci(0),jf) + a212(0)~v- l .

Since th&olution of a difference equation of the type

we obtain together with the starting value >JO1) = Nl(0) (which is a consequence of 00') = 0) and the -representation (3.36) for 4:') :

(3.49) "(" + l ) Q - Y - l N o ( 0 ) ,L* < =Q-YNl(0)+vQ-va2,2(O)N~(O)-wc~(0) ,. 2:2ce kjil ,, *-

+ a212(0)No(O)~ - UQ-~C~(O)NO(O)"(~'~)] 2 Q-v - N1(rY) . -."

By linear combinatiah of the equations in (3;46) (where the second equation is rnultiplied by s) ggd-yith the definitiion 4%: 33j

G1 2(0) (1) (3.50) $1 := [L1) + (o) ij,,

Together with the representations (3.40) for ,(P) and (3.36) for ij?) we abtain the general solution of (3.51):

With

W& $#& up with the following representation for C") (due to (3.45), (3.50) and using the represen- 1

tation (3.49) for G?) ) : "'12(O) >r(l) - Mi (TV) = = ("1 - M~(T,,) = (L1) - -- .wc2 (0)

4

. - - B

Since l i n ~ - + v ~ ~ - ~ ' = 1im Mi(rV) = 0, the condition lim C(') = 0 requires v4QO ' V 4 0 0 V H O D

Hence F(" is fixed; (3.54) and (3.55) yield for U = 0 (due to Q = 1 + wc2(0), 1 - Q-O = 0):

Ci1) = -Ml[O) - - a112(0) (1 + wC2(O))N1(O)+ wc2 (0) I

+ a112(0)c1(o) l ( l $ wc2 (0)) ~ ~ ( 0 ) + a.l~?(0)al~l(o) + wc2 (o))~N~(o)- cz"(0) ci(0) w2 -- .

. ,

(3.56~) --- b(o) l No(0) + a1'2(o)c'(0) 1 ( 1 + wc2(0))No(0)- C2 (0) W ci(0) w 2

ä112(o) L( l+ wc2(O))No(0) - a112(o)a2j2(o) + wc2(0))No(0)- I c2(0) W cZ(0) w2

.? a:*2to) 1 - , - ( I+ w ~ z ( 0 ) ) ~ No(0) . 4 0 ) W2

&&kk&Q%&,and after sorne.si~ple. maqipulations and rearrangement ( (1 + ~ c ~ ( 0 ) ) ~ ~ is split into (J wedo)$ + w~ (0) (-1 + wc2 (0))) this results in:

20

.

i .,.l , - ; ' , . .

J<.. , 2: .+.!E . > T L ' \ ' o b * . W'-'(1 + wc2 (0))noli (O)+

7 ~ ? . ~ > j < > $ j p - . : : b ~ , 3 ~ . . , I *

+ 2 [a1,2 (O&r .I (0) ---- ai,1 (0) "'(O)] W ~ - ~ ( I + wc2(~))no, ; (~)-

i=l c2 (0) c2(0) c2(0) ' ;.: ' 4 * 1

- C -w'-3noli(0) . hr&ic& b; . . !=1

::L%- $+-

Pfow we rnay write (similarly as in (3.42) above)

bz&ause n1,o 0 by dcfinition (in accordance with [1,(4.72)]) and nlP4(O) = 0 due to the ~ ~ - ~ r o ~ e r t ~ (cf. [I, (4.93b)I). Furthermore,

-Q p $ . F . >+*" . . . * . 4

.(3,58):: ,.E[* *I ~ ' - ~ ( 1 + wc2 (O))no,i(O) = C [*; -1 wi-'(not*- l(0) + c2 (O)no,;-2 (0)) , .. ~

:9* C: . C , - . -

i=l i=l \ . ~ h - < ? f , ~ 7 - , C c X x r A r . .

2 >, , ' -r.9 e C -L'-'no,;(~) W) =.C --w'-4no,i-l(~) b(O) ; X $ ,

i=l cefQ) T c ~ ( 0 ) ?2&i::- :.

hold due to definition [1,(4.72)] and due to the so-property (cf. [1,(4.93b)]). Henee we obtain , -, ::*.

*4-2\

1 c3.%3) arid c3.59) show that, indeed,

due to our argumentation in Subsection 3.2. I.e., p, (and therefore Ru = Tupu) contains no smooth

~bt!&~~~olution cornponents of reduced order O(hi), i < 5. (Moreover, yj3) 0 at 0(h6)-level.) Thus only inner solution terms are present at the level of reduced powers of h. From the above qqggeaegt@kqs ( cf. (3.361, (3.40), (3.49), (3.54)) and the fact that

(3.61) Mi(0)=O(w-I), N i ( O ) = o ( ~ - ' ) , l = O , l , ... , ::>fi?$

whidi is a,consequence of the ~ ~ - ~ r o ~ e r t ~ , it follows that the leading inner solution terms are of the form ,

, . . .

. BI.&T~+(@) +-\-W,.arrS.- , . :denotes - some fgeneric) polynomial of degree 5 i in V whose coefficients may depend k

iq pse a ?+,* harmless .,k9-v 5 way on W , i.e. on negative poyers of W or on terms like (9) or ( g) whi*, are E r br '%

i t ~ ~ f b . . ~ ~ ~ b o ~ , n d e d for $ Ch. p;, (T) a gqeeric polynomial of degree $ i in T whose eoeffieients p ss=a,s 1 1' 3

are aff4t&a with &tziln (positive as well as negative) powers of w. *) Note that due to the EO-

property t&:conatant coefficienbs P*;, (0) of each pi,, (T) within (3.62a,b) satisfy (cf. (3.36), (3.40), (3.49), (3.54) an& (3.61)): . . "3 :

(3 .62~) pi;,(0) = O(w-I)' for E C h .

(3.62) shows that the inner solution components consist of two types of decaying terms: There are "discr6tely eiponentially" decziying terms of the type waip;. (v)Q-, and exponentially decaying t ~ ~ ~ ~ # i ; ~ It is easy to see that p; (v)Q-". iq,uniforrnly bounded. (~,)e-~2(O)'- is uniformly bounded, too, for 911 v 2 '1 in spite of the presence of positive powers of W (cf. [1,(3.5)]). Since the decaying behavi~ur of ~ ; ( : V ) Q - ~ is weaker than that of piiw the damping

E of the order reduction effects with-increasing v is characterized by the discretely exponentially I &&Y&g 6 e m l ; i ; ;

* t + * ; d 4 L ;. L . 3 - , L . '

- . T $$.$*E F j ;,. , .. . >

*) $~d@&@~.tiiS"polymrnials psiW (T) of degree 3 and pc, (T) of degree 5 appear in (3.6%) and (3.62b), resp., f~llows easily from the singular perturbation analysis of the V.E.'s of Part 1; nevertheless, ,

the degree of these polynomials is irrelevant for our purpose.

b+ , b r

w-l(v - I ) * Q - ( V - ~ )

. %

(3.66b) W i r l i i e c 2 (0)~"- I 9 3 2 1 2

,&*?+*-. *$. *" - "(3,*3@4 ' w-le--t2(0)r.-'

g * % , C

(j) (j) &66) originate from Mi(rv) and Nt(r,) (via IZ: and indirectly via Ev , QV ,j = 0) 1); ~&:e~Oqtj.ctbat. w-'. appears-in (3.66~) is a consequence of the e"-property (cf. ,(3.62c)) and of the @@tt&fi*sd'%~f~~, and qol within (3.64b) are not affected with W.

respective solution compoBents in read as follows. From (3.65~~))

Y

, ~ k - l - ~ k ~ ~ - k = W - ~ W Q - ~ ~ Q - ~ C , k i = W - l p i + l ( y ) ~ - ~ , k = l

Y

iti uniformly bounded and C ki = F ~ + ~ ( V ) , where again p i + l ( ~ ) denotes a poly- k= 1

&~@a1-:ih;v.&f-de~ree 5 i + 1. (The second term in (3.65a) yields even a factor wm2 .) From

V

) C ~ ~ - - ' - ~ w - ' (k - l ) ' ~ - ( ~ - ' ) = W - ~ Q - Y (k - lli = (V)Q-Y+ k= l k = l

see thst the discretely exponentially deeaying terrns in (origipating from multiplied by a factor W-I .

Y

(O) rk = w i + j ~ - l ~ - u C kj(Qe-wc2(0) = k = l

Ik \

& ~ + Y Q - ~ Q - Y ( Q ~ - w c z ( ~ ) ) [p j (~ ) (~e-wc2(o) )Y + C] =

-wc2(o)r. + eonst. . Q-V].

- d@8@6f6*esdki~ialthat lcvl = 1(1+ wcz (0))/&~2(') ( 5 const. < 1 (due to our assumption r 5 Ch) z@&oh&h_a,t .f3,63d) iholds; notice further that w'+j+1e-wc2(0) is uniformly. bounded.

, *--iC. Ci-& , .

V-1 Q k - l - ( V - l ) , i T ~ e - ~ s ( 0 ) ~ k =

&=I

24

6anb&e&roated fin the sime way as (3i68a). @g:"mr$&b;q we Lave

. > t y ; p "', -- -!U t , Y

% C Q k - l - ~ W - l e - ~ z ( 0 ) ~ ~ - I = W - l ~ - ~ [ge-wcz(0) k-1 = *&$,$L --" 2", '$LLt1 : k t l

I (3 B8c)

* 4.- -- ~-- '9 - -~ + W - l ~ - " [ ~ ~ - w c ~ ( O ) ] k .

k= l *$ ; $l $ .,=> , L

Here, the Q-Y-term has again a factor U-'; the second term is analogous to (3.68a,b). *gcyy .$" &umrnarafng, we See that the structure of t#) (in particular w.r.t. powers of U) is essentially the

~ L ~ ~ ~ q $ i ~ & qf (cf. (3.62)) :. The terrns decbying like V'Q-" haue. always a factor W-I ; the same is true for the terms T : ~ - ~ ~ ( O ) " . Any powers of W appearing with ~ j e - ~ z ( ~ ) ~ u , j 2 1, do q@@%@atte.

component €,L2) is treated as follows: As usual, the starting value er) is chosen such 0. halogous consid&aGons as for above show th& (L2,) h w cysentially the

, ,

S$mew&udiire as t#); rnoreover, the etwting value C:') turns out to be o(w-l ) . *) (This can h7wkwn. byc 6sirxgg (8.68) with a = Q -' and a = e-"~z(~); resp., and agai. taking aceount of the ~ F A H R ~ ~ ~ Y , J: .

;,*-2

@&trAicierrtsj~fof:h.' : \

3:&&j 7 we have @i3' z 0. xL3' has the starting value (cf. (3.13)):

- &j$(g:-u:,: :.+ . ? :

(@$@$i,: &S) = 0.a. ($2) = - 6 4 2 ) o(',)-'); r*g& "&'$>%X . ,

& ~ ~ ~ t h e s o l n b i o n of a simple difference equation with srnooth coefficient functions **) and a smooth inhqrnogeneous term k c i = 0 (1). By the usual classical stability estimates we immediately &&g@& ' .

($gZ@)..; = O(1). - P .

For ehe so1,tt~iCon of equation (3.28b) it can be shown by the Same inductive arguments as

fb%~vf) S ~ P Y ~ ~ that it has the sgrne structure as j 5 2 (note that, according to (3.13b), , "(3)

r / p = a , Y 0 =o). *

&g$ f.3bt~' ,:&3 : ,. "

Remainder term py) of the h - ezpansion of P, : i I

TQe ?rnain(ler term plR) (cf. (3.14~)) in the expansion (3.11) for p, satisfies the difference equation (9.94; %dqidiiig g I + ' .. to (?L131 the starting value is a t O(h5)-level.

. 3 8

*) Re~all- that we are discussing the case q = 4, i.e. the O(hq+l) = O(h6) remainder term. If we had 3' .aa iHf&~ k ~ ~ d b i d d e r %er& of higher order (q > 4), i.e. if we had discussed more'(and not only ehe

,inner golutiön terms of the ~ ; ( t ) ' s (i 5 p) in Part 1 and if we had expanded all up t o a 'the&k could be shown that not only (r) = O(U-l) holds, but that, again, (f) = O <a$jJ' '- . Lt$2) G. 0.

U-POO

&n~&n@bions&,,~f t.) in (8,28a) depend on,:h, but in a harmless wsy (cf. (2 .B) ,(3.7b)), ~ ~ h ~ ~ m ~ ~ o b i e ~ ~ %fdr the stability est imate.

Iw&cde~ato-estimate p y ) in the spirit of the B-theory (cf. for instance [3]), we introduce the nonlinear function

which is one-sided Lipschitz continuous with the (moderatel~ sized) one-sided Lipschitz constant

=dognorm(A,) + Lv

(3.73) At(0) = 0.

tli respect to T( t ) we can assume that 8, is bounded:

(&Ji'j#) k.j11& 1-8 for d l U .

from the collection of all terms which have not yet all that, when equating coefficients of h6, we ornitted

(cf. (3.28b)) such that the respective' h5-terms noiv r order). Recall further that 6':) originates from = due to the re-scaling of the second equation.

Hence all terms in 6LR) have a factor h5 in common. Hotuever, 6LR) depends also on various powers of the parameter w = 4. For E < Ch, positive powers of w are critical. But positive powers of W

%&@%%~")1in 6LR) anly in conjunction with T ; ~ - ~ ~ ( O ) " W , j 2 1. *) This follows from the fact that all teims (originating from the recursively knbwn quantities (P', t$)) which decay like. U'Q-Y or

* + , %,< - &%&)e*%io)'~ contain a factor W-' (cf. the above-discussion of the structure of the (F kd

such that after re-scaling of the second component, i.e. after multiplication by f , these terms are qt&fF%gtgd,with -F positve powers of U. Hence it is guaranteed that

'I'be B-convergence estimate for is now straightforward: Consiiier the step

(R) -[pLR) - p,] = ~ : ( p ; ~ ) ) + 8,p,-1 + 6LR),

@d:&he wialogous step of the homogenous difference equation:

.** d&$j$.j& -*?,~y$L!<; *2;.g; , L .

d@5t,$$f$igfpf,(.3-71) f r ~ m (3.76).,

ints. This is important because B. nce estimates provide no insight into damping effects such that it is indeed required that

ombgeneity has the full order for all V.

26

aa$ aISGREj,TE SINGULAR PERTURBATION ANALYSIS: @:&,E;~C;&SE ~h > k ,

- i*;; $*&*-;L: : ,

, - @qs,@ktgf.?$ $ 2 .. : 3. P

&&&e&$cjh 3 ?b.qye;,-:the assumption c 5 C h was essential. In the lower part of the E - h - planen, i.e. fad & @g&-&aomg mbderate constant), the various expansions w.r.t. powers of h (cf. for instance 4 & . & ) $ ~ ~ ~ ~ t i r e . a o ~ b l e : For h « c negative powers of w = $ become arbitrarily large and are not $j~ps&ss&&d~~at'leapt .not at t h e first grid points) by the factors e-~z(O)'~ = e-c2(o)wv ~s 1 within $ham$$(k) fandJ (T,).

yJ$*@:&$$y2*:i 1 - , . & vatje ; the fgllowing, G: . let us have a look at the E - h - plane which? divides into several

su omains representing the various cases considered so far (cf. Fig. 1).

C - 0

/

ga2 &ixgv2gfh x C T B

F& *&s$+ai&, "

.

.

azp9! % -L-:., ; - - , .. C

$$"Z.$f -,P' .*,, - >-

- - 0

'-Si - t < - B A _ :;;L .<) .. .

&&-&*&-k g>

1 A i

Fig. 1

%+ il i . : rc.-~A.i . subdomain of the E - h - plane for which the problem is not stiff (classical

g* i .:F- : - cäse: E is not a small parameter) 5 ,~

6, E' . - - ,, - "*B... subdomain of the E - h - plane which is covered by Part 1 of thispaper ( E ,: < .

&c*j&L$ + -

is so small that the inhomogeneity b, - C, of ($4) is a t o(hq+l)-level)

&&g5-e-I -.C:.. subdomain of the c h - for which E 5 C h (covered by Section 3)

-haue characterized (the clacrsical case by " E not small" . This appears. not t? - be nwe,with usual de f in i t i q .of stiffness .distinguishing between " non-stiff" ,ad "&F .via ".L not small" , resp. (L is fhe conventional Lipschite constant, i.e.. L = ~ ( i ) ) . For

$$gitioiiiofstiffness, the case h Cs or 5 e h h. L 5 C would indeed- be the. "dassical case" , -~~~@A$$&otJd:8xpect that no further analysis has to be done. For the present purpose, h o ~ e v e r ~$&$&r$~mg&$gjabe of R, in the Bsense), we have to go back to our definition of "classical casen.

assical and Btype a-priori convergence estimates for the implicit

(classical estimate) , , rn 5 0 (Bestimate).

h d for IIzN(t)ll, t E [O,T].) From (4.1) it can be Seen that, even for hL << 1, the - 1)/L is much larger than (emtw - l ) /m (if L >> 0 and rn is of moderate size). The

; " - i; p;srises for sstiqiates of R,. Thus we See that we must indeed analyze the case of a

&:SrC& to cover the whole E - h - plane. ,

28

- -

&mentioped,atqthe beginning of this Section, an expansion in powers of h is useless for h 5 CE. It will turn out that a discrete singular perturbation expansion w.r.t. the parameter E is appropriate in this Gase, ~li t .%e-have to be careful: Recall that, due to the nonlinearity of the remainder term g.!&+$ gi*' .--. . equation (1.4),Iriegative po*ers of r must be avoided (cf. the remark at the end of Section 1). To this end it is prompting to strive for srnooth solutions of the variational equations. This necessitates t~Ei~@iedhtrst~t'8-f:fd~(;2:15~] and [1,(2.17)], starting values for the ei(t) have to be chosen such that no Jpner solution components appear. In the rest of this Section we will sketch an analysis based on $J$~tea~(4.ii!I&out göing into all of the details). It will turn out that this particular approach works &~"c"ce~sfull~ foy h 5 CE but fails for r < Ch. Thus, both types of singular perturbation analysis - with an ansatz in powers of h as presented in Section 3 und with an ansatz in powers of e based on

F 4.1 ~rno&h - 7 :~olutions of the Variational Equations

b%fpi: ",' smooth solutions of the V.E.'s, special starting values have to be chosen for the Y"'B&(i). Due to the identity gj,+ ;' ' E

C (42a) $Q,- ~ ( 0 ) = hei(0) + . . . + hqeq(0) + & @' .V*- 14.9 gt, [ k s l l &hat - ~ ( 0 ) is the accumqlated error from the preceding integrstion intervals), the starting

.O~hq*i)-level g&ygq isichieved with increasing V. A proof of this dsmping property will be sketched in

,.. Wf3gd 4.2. the rest-of the present Subsection we give a precise characterization of smooth gg .," C%,p d ~ i u t i o n s in the Qingular perturbation context. "2

- C &yy"-:L "

hn solutions we mean solutions of the V.E.'s for which no inner solution components are *,%,. ' tt; *-G $ $ ~ e # t o a certain e '-level and for which the starting values for the outer solution components

"..'* I, Pqi(ty (I * 0,. . . ,,k -.I) and forsall further components (at rL-level, 1 2 k) are O(1). It is W , ? - - + wious &bat k can be chosen such that the respective terms appearing in the inhomogeneity b, - C,

~ f a ( j : 4 ) * &-U are p(hq+') (derivatives w.r.t. t up to the k-th order of inner solution terms at ek-level

vg;aea), &lQ&i@pqiei(t) in the above sense can easily be constructed by a slight modification of the

F ' U ~ GVamcertain ek-level. For these 'starting values we make the ansatz &&%.,*W * z

k&p&!y * < - - ' <

&@$Hat we a&n~choose the Q,i tarbitrarily (of Course O(l)), but that the terms in the second @$&,ar~68ben determined recursively (to define a smooth solution ei (t)), i.e. y~ , i is fixed by

asily be Seen by making the ansatz *)

ance [1,(4.13)] or [1,(4.15)]):

a l . z ( t )x , i ( t ) + ~ l , i ( t ) ,

t ) , 81-l,;(t) depending recursively on the smooth func-

[a2,l(O)xi-l,i(o) + 0 2 , ) (0)YI-I,*(()) + 81- l , i ( O ) ] =

--

. . ( t ) can bechosen arbitrarily at O(1) - level:

O ( h ) , y t ) = ~ ( h ) , i = 1 , ..., k - I ,

ysis for p, to establish that the outer

( t ) ) ; the starting values in the expansion are set in such a way that ursiveli turn out to vanish up to to the ~~-1evel . Course O(h) ; the accumulated error from the preceding intervals

onvergence; hence po = 0 (h),.

q + i ) . Analogously as at the

ere ezists a solution pLR) of eqwtion (4.20) w t h a starting v a k e prds ?tR) with increasing v such that

e following: At the endpoint t# = T of the actual ); now the differente equation (4.20) is solved in the homogenous kerm A:~), i.e. if we would solve (4.21)

&di~&e&8n, we would reproduce the 0 (~~hq+')'-solution ?F). The dffects of AL^), tui:lj&tion of (4.21) with a very epecial decaying behaviour (i.e. an 'increasirig -1 . .: -r O ) , can now easily be estimated by standard stability aigum&hts; due S e *f AY) it C& 'be guaianteed that the effects of this perturbation increase

eway (in spite of IIAvll = 0 ( + ) ) such that we end up with a s t ~ i i n g value W@): n$b6:thisz &r&inentation it *-essential that we häve modified the uijual Siirgulai

in tbe sense of a more precise description of th'e decaying behaviour of the (4.16) above), i.e. that we have not expanded the data functions cz (t), . . .

%I%# $~bQ;~&h&;4b;e,pguations defining the inner solution terrns; only this enables a sufticiently (R) 4@&p&s,t i&.a&e ~f:&he~effects. of .Av .

Ehg &gcit&~c, -*f plR) with piRR) = O(skh) which tends to O[ckhq+*) with

( $: ) for (4.7.) which satisfies (4.7~). Since ,

es a t the ek-ievel in the s- O(1)-level (not only in the first but also'in the

ible with smoofh solutions

. ~ ~ ~ ~ o ~ l @ d ~ ~ " z r r snalysis ior h < CE. *,g$&->f - C,> &* $&-&$. 2% I < . .

"5. -CQNCLUSION

and h 5 CE of the E - h - plane are covered Course of major practical rglevance. Only if acy ;&uirements are so strqng . that. stepsizes

lied. For s t 8 problenk, (E It rnight be expected. that structure in f he transients,

,qf %original problem is not smooth and therefore very small stepsizes h (and .k,e used to p,atisfy. certain accuracy requirements.

* I

&~Se&tin"-4~does :not -apply here: In the transient phase, smooth solutions e i ( t ) of the uations~do not existain general. *) niis seems, rrt first sight, to destroy our inductive .

,E.% contain certain derivatives of t;(t), which cannot be assumed to uations defining the inner

34

.&F&*$&@&.>< -:5 - zii&:- , > aaimen%gdr~%P13:stmting values: Recall that we have throughout assumed that the starting values e i ( t ) are of the form

Tx;7&*z.

HL&sirv at the end of that subinterval, such that there remains no term at eo-level in the

-,";-. 5 e r m o c - ?$,:@$er scheme, evaluated at a point t = er, shows a factor r due to = O(E). Thus

'.-%

?i&y.:y$ L

B&fied t o assurne that all solutions of the V.E.'s are at O(r) - level immediately iafter the ;\2>:$T;L:K:*y W,$%" " z;yl :.,*c, + . ; . $$$@f.g@xii this shows thst (5.1) is always justified.

l&2$so be noted &hat our recursion over subintervals with constant stepsizes (with starting

fterent considerations (Section 3 in conjunction with 111 or Section 4, resp.) have to be

ding Section, briefly remark another point concerning nonequidistant grids: e the assumption of coherent grid sequeaces and the question arises whether

strictive. To our opinion this is not the case as the following arguments

ences can be perfectly adjusted to the local smoothness of the solution data. Only in the transients, where the local smoothness of the solution it might happen that, if for a certain accuracy requirement a certain grid

al (w.r.t. efficiency), none of the coherent refinements of that grid would be ch stronger accuracy requirements.

lar grid which is generated by a certain code with a certain stepsize control, ticular problem, can be interpreted as a member of a coherent grid sequence coarsest grid defining that sequence) .

&$;@rp$ent paper we have extensively discussed the asymptotic expansion of the discretization ?!&he,-implicit Euler method. The question arises whether similar results can also be proved

her methods. In a forthcoming paper we will discuss asymptotic error expansions for the s:*.dpoint rule and the implicit trapezoidal rule, trying to extend the results of Dahlquist !aberg 121. In contrast to the strongly stable implicit Euler scheme, there is no damping

ie'.\i$fference equations for the midpoint and trapezoidal rules. If, therefore, the order of the !%&k term breaks down at the first grid points, no damping effects are to be expected; hence

P~bel order remains present in the whole integration interval. There is, however, some hope z$$fete dingular perturbation techniques can be applied to establish a eystemetically oscillating B)- ~f the dominant components of the remainder term. Therefore we expect that certain &&l.-procedures for error estimation, stepsize control, convergence acceleration etc. can be

@@'is-whether U%csq- the full asymptotic expansion exists in the strongly stiff case r < h (as it is @ f ~ r ,% ir the implicit Euler scheme - cf. [I]). Simple models show that this is not the case in

o d y for certain classes of problems which we will try to characterize. The trapezoidal kf&&ms to be more "robust' w.r.t. this point.

rnptotic error ezpansions for stif equations, to SIAM J. Numer. Anal.

t one-step nethods for stifl diferentid eqw- itute of Technology, Rep. TRITA-NA-7302,

Ueberhuber, The concept of B-conuergence, SIAM J . Numer.

ngular perturbations, Academic Press, New York - Lon-

8 ,