ASYMPTOTIC ERROR EXPANSIONS FOR STIFF EQUATIONS. PART 1: THE STRONGLY STIFF CASE

W .AUZINGER, R.FRANK and F.MACSEK *)

Abstract. In this paper we derive an asymptotic expansion for the global error of the implicit Euler scheme applied to stiff nonlinear systems of ordinary differential equations. Part 1 is devoted to the strongly stiff case. It is shown that in strongly stiff situations a full asymptotic expansion exists at all grid points. Our analysis is based on singular perturbation techniques.

1. INTRODUCTION

During the last years there has been a considerable Progress in the theory of discretizations of nonlinear stiff initial value problerns (IVP7s). For a large class of methods global error bounds have been derived which are independent of the Lipschitz constant L of the right hand side of the differential equation; they only depend on the one-sided Lipschitz constant m and on some other moderately sized quantities characterizing the given stiff problem. (Concerning the relevant notions as for instance " one-sided Lipschitz- continuity" ,"G-stability" , " B-stabilityn , " B-convergence" ,..., See the respective literature, e.g. [2], [3], [ 5 ] , [6], [7], [8].)

It is natural, as a next goal, to strive for results concerning the structure of the global discretization error. I.e., we are aiming at asymptotic error expansions of the form

(1.1) - z(tU) = hPep(t,)+ . . .+ hqeq(tu)+ R,,

where C, denotes the numerical approximation to the exact solution z(t) of the given IVP at the grid tu , obtained by a method of order p. h is the mesh size of the grid used. The functions ei(t) are solutions of the so-called "variational equationsn. For any given method the derivation of these variational equations is a purely formal procedure based on simple Taylor expansions and arrangement in powers of h. The crucial point is to estimate the remainder term Ru , i.e. to show

I (1.2) Ru = O(hP+l).

For lar,ge classes of methods, estimates of this type are well known in the "classical sense" where I R, depends on the Lipschitz constant L. For stiff problems, the essential questions arise whether

I i) the functions e,(t) remain bounded independently of the stiffness,

ii) (1.2) remains valid in the B-sense, i.e. the 0-constant is only allowed to depend on the one-sided Lipschitz constant m and some other, moderately sized parameters ~harac ter iz in~ the underlying problem.

To our knowledge there exist three papers dealing with asymptotic error expansions for stiff s y r tems.In particular, the implicit midpoint rule is discussed by Dahlquist and Lindberg i4! and by

1 Veldhuizen [11] . The implicit trapezoidal rule and a eertain semi-implicit two-step method are considered in [4] and in Bader, Deuflhard [I] , respectively. The results presented in these papers,

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

however, are not rigorous in our sense, i.e. the validity of i) and ii) is not verified for the respective methods. In particular, the estimates for the remainder terms presented in [I] depend in an implicit way on the Lipschitz constant L.

In our paper the questions i) and ii) above will be discussed for the simple implicit Euler scheme. We will show that for strongly stiff situations a full asymptotic expansion in the sense of i), ii) exists a t all grid points. In part 2 of our paper we will analyze the mildly stiff case, where order defects of the remainder term inevitably occur at the first grid points. Remarkably enough, it will turn out that the full order reappears a t the later grid points.

The present paper is subdivided into 4 Sections. In Section 2 we derive the variational equations and a difference equation for the remainder term. All this material is of Course state of the art; we have included it for convenience and to introduce some denotation to be used later. In Section 3 we study a simple scalar model, demonstrating some of the effects which are to be expected in the general case. A general theory based on singular Perturbation techniques is presented in Section 4.

2. THE VARIATIONAL EQUATIONS AND A DIFFERENCE EQUATION FOR THE REMAINDER TERM

Consider the IVP

with the exact solution t ( t ) . The implicit Euler discretization of (2.1):

yields approximations qu for t( t ,) at the grid points tu = vh.

Let zh (t) be a smooth function interpolating the C,. We make the ansatz

For the moment we restrict our considerations to an expansion up to h2 and a 0 ( h 3 ) - remainder term. The extension to an arbitrary order is obvious.

Since zh ( tu ) = qu , (2.2) can be written as

To establish the asymptotic expansion for the global error we insert the ansatz (2.3) into (2.4). Taylor expansion and rearrangement in powers of h will y'ield the desired equations.

Consider the Taylor expansions

+(("J)*'"J)j - ("J)/Z = Y = (("J)"z ' "J)! - ((y - "J)"z - ("J)"z)- = 0 1

U! sqnsal y 30 SJ~MO~ U! Edu!%ue~lea~ pue (P'z) ogu~ (L-z) pue (9-Z) Cdu!q~asu~

. [("t)Zazy + ("7)Tql . qz(O - l)((("J)zaz~ + ("J) 'ay)o + ("J)Z '"J)"""! Y+ C Iz T

+[("J) Zaz~ + (nJ)~~]((nJ)z '"J)IEl + (("J)z'"J)!=

= (("J) ZazY + ("2)Tay + ("J)" "J)!

0

.q((A~)~~o + ("~)za~y + (A~)Tay + (nJ)2<nJ)fi~ J = : ((A~)~~)!

T (?L.z)

'("J)". ' (("J)".)i + (("J)ZazY + ("1)Tay + ("J)z'"~)/ =

= (("J)". + ("l)ZazY + ("J)Tay + ("l)zg"J)j = (("?)"~'"7)/

oqu! papuedxa s! (("J) "z ' "J)!

.((Y - "J)Y. - ("J)".)! + 1 + Z'"I~ - ("J) Z,a Z~ +

z +""I - ("?)Na- - ("J)~Y + t

zY Y 9 z Y +O*"I~ - ("J)///z- + ("J)//zV - ("J)/z = ((Y - "J)"z - ("i)Yz);

zY

where

I (2.9) C, := - h I u , o - Iv,l - hI,,z,

and 6 , is the collection of all terms of order greater or equal to 3 from (2.7b).

1 Since z(t) is the solution of the original problem (2.1)) (2.8) is satisfied if el (t), e2 (t) are solutions of the linear differential equations

I (2.11) 4 ( t ) = fy (t,z(t))e,(t) - :zut(t) + ie:(t) + fyy(t, z(t))e:(t),

and if the discrete remainder term Ru = rh (t, ) satisfies the nonlinear difference equation

I (2.10), (2.11) are called variational equations. They are defined in a recursive way: The inhomoge-

I nous term in (2.11) depends on ei (t). (In general, the k-th variational equation depends on d l e,(t),i < k.) Similarly, the b, and C, in the remainder term equation (2.12) are defined in terms of the solutions of'the variational equations.

I The starting values of the ek(t) and of Ru remain to be fixed. For t = t,, (2.3) reads

(2.13) F - ~ ( t , ) = hel(t,) + h2e2(t,) + R,.

I In particular for U = 0 ,

(2.14) 0 = hel(0) + h2 e2 (0) + &J,

I since $0 = ~ ( 0 ) . A natural choice to satisfy (2.14) is

(2.15) e1(0)=e2(O)= & = 0 .

I Up to now we have only considered equidistant grids. Of Course, nonequidistant grids are unavoid- able in the context of stiff problerns. In the transients it is necessary to use a very small stepsize to obtain a good level of accuracy. When the transients have died away, large stepsizes (adjusted to

( the smoothness of the solution) can be used. We assume that our particular nonequidistant grid is a rnember of a so-called coherent grid sequence in the sense of Stetter [1.0], in which the subintervals with constant stepsize are kept fixed during the asymptotic process of the stepsize refinement; i.e., the points where stepsize changes occur remain fixed. In this sense, the interval [O,T] has to be interpreted as one of these subintervals with constant stepsize. Thus, $0 f '(0) must be admitted in contrast to (2.2). The quantity go - z(0) is the accumulated error from the preceding intervals.

Therefore, (2.14) has to be replaced by

where the left hand side is the accumulated global error a t t = 0 ; we have assumed that h is the stepsize used in the preceding interval, el(0),& (01 are the solution values of the variational

I equations at the endpoint t = 0 of that interval, and RN is the remainder term at this point. A natural choice for the new starting values would be *)

- 2

I (2.17) e1 (0) = $;,(0), e2 (0) = ( t ) e2 (O), & = RN. *) Note that for coherent grid sequences the relation between the different stepsizes remains fixed

- i during the stepsize refinement, i.e. ( t ) is always constant, and so the starting values e;(O) (and consequently the quantities ei (t) ) are independent of the stepsize parameter.

The nonlinear difference equation (2.12) for Rv is of the Same type as the implicit Euler equation (2.2), and so it is obvious that R, can be estimated in the Spirit of the B-theory (cf. Frank, Schneid, Uberhuber [6]): Using the well-known inequality

(2.18) II[I - h i v ( ~ v ) ~ - l ~ ~ 2 &,

(where m denotes the onesided Lipschitz constant of f ), it can easily be shown that *)

This would yield the desired result R, = O(h4") (cf. (1.2) and question ii) of the introduction) if

(2.20) b, = o(hq+l), C, = o(h4+l).

By definition, the b, and C, depend on certain derivatives of the ek (t) , k 1 q. The ek (t) are solutions of the variational equations which are of Course stiff since their Jacobians coincide with that of the original problem. Thus it must be expected that the ek (t) (with the starting values (2.15) or (2.17)) are no smooth solutions of the stiff differential equations (2.10), (2.11), resp.; i.e. their derivatives are influenced by powers of the Lipschitz constant L. Our aim is to investigate the consequences of this phenomenon in a rigorous way. In part 1 of our paper we discuss the strongly stiff case. It will turn out that (2.20) holds for all grid points v 1 1. In the case of mild stiffness, which will be analyzed in part 2, order reductions will be observed in (2.20) at the first grid points.

In [I], Bader and Deuflhard - discussing a serni-implicit two-step method - derive estimates for the remainder term which are analogous to (2.19); but they do not analyze the crucial point, namely if a relation Like (2.20) really holds. For the implicit midpoint and trapezoidal schemes, Dahlquist and Lindberg [4] use another approach: In order to obtain smooth solutions of the variational equations, the starting value is modified after each change of the stepsize, which causes considerable difficulties in practice.

1 J. A SCALAR MODEL

In this Section we shall consider a simple model demonstrating the effects which are to be expected. To study the typical behaviour of the functions ei(t), we look at the scalar IYP

( with the Solution r ( t ) = e-'. The first and second variational equations (cf. (2.10), (2.11)) and their solutions are

1 ei(t) = h l ( t ) +ie-t, el(0) = 0 ,

(3.2) 1 - 1 -t

= 2(h + 1) 2(X+ l ) e

*) For m = 0 the quantity &[emtv - 1) in (2.19) is defined by lim &[emtv - 11 = t,. m-0

-squ!od p!iP 112 qe sieadde (J)% qy JO 4 iapio IInj aqq snq~ -- - "Z'T = n '"3 = qe q~a3a ou aAeq suriaq ~ea!q!i:, aqq laqq queup~~op 0s 6! (du!durop p!quauodxa aqq (1 « yIy1) aseagqs AlPuoiqs aqq U! 'qa~ '(aqoupuop qou saop „ya JO ialihod (du!durep aqq (du01 se) squ!od p!iP qs~y aqq qe z oq paanpai s! (3)qaqy JO Y iapio aqq (T - Y [Y I) ase3 B!V AIPIW q

' -C+? (I IY I) JO ssaupapunoq uioJ!un aqq oq anp

adAq aqq JO miaq su!equo3 J(~)qal.,y 'yn = "3 squ!od p!iP aqq qv .iaPuoiqs Bu!qqa(d s! ,ya JO Ae3ap p!quauodxa aqq (U!

: Z < Y JOJ PPunoqun aw(, ya C (X) z -qY "'OJ

aqq JO miaq Pu!u!equoa) (I)% aqq 'snq,~ -7- = ,I qe pauyqe s! C (;) ~nnnmm aq,L -0 7 I 101

y U! dpioj!un papunoq sie I ,?!(X) I suo!73unJ aqq asn=aaq ,ya fi uo!wqou aqq pasn a~eq a~

:(Y JO wamod Bu!peal) miaq 1~3!q!i3 qsou aqq -q.i.~ sno!Aeqaq p~!qeuraqsds aqq uo daains e aal2 aA aIqeq ~U!MOI~OJ agq ul Pu!a~oau! miaq JO no!qo!quaiapp Aq pasne3 'ieadde y 10 aa~od aiou pur! aiom (I)% suo!qnlos qaql U! se 1Iah se suoyenba puo!qe!iea aqq JO suxiaq snouaPouroqn! aqq q

. , ,(T+ Y)Z~ - ,($Y] (1 + + + Y)ZI = (3) „ 1 - T-YZ 7y Y T -YZ

(1'2) < a '0 = (0) Za 1-

(1 + Y)ZI + (' + + (I) gay = (I) P T - YZ zY

When investigating the remainder term (cf. (2.12)) we have to look at C, which consists of terms of the form hi-I (cf. (2.9)). For I > 1, I,,,( is the remainder term of a Taylor expansion for el (t). The integral form of the I,,l shows that C, is influenced by the ~ ( ( X I ' - ~ ) - ~ e a k s of the functions el (t), the effect of which has to be studied carefully. *)

In the nonstiff case (X moderate) it is obvious that the respective integral remainder terms h 1 - l i 1 have the desired order. In the stiff case we separately consider the hl-I Iu,l-tenns originating from

1 ( 36a )k j r "

1 (cf: Table 1). For U = 1, the (3.6a)-terms yield (after some simple manipulations) hL-l Iu,i-terms of the form

(3.7a) hl-l . (X + linear combination of ( ~ h ) ~ e ~ ~ ) ,

whereas the (3.6b)-terms result in terms of the form

(3.7b) h . (linear combination of (Xh)' exh ) .

I For & << h (strongly stiff case) the terms (3.7) are irrelevant since they tend to 0 for X + -m,

and therefore the other O(hq+')-terms of hl-I IUIi (not originating from t&t or ~ ~ - ~ ( A t ) j e x ~ ) dorninate.

1 The situation just described is typical: There are certain quantities which depend on h and on a paramiter X in such a way that they tend to 0 as X -* -4 when h is kept &ed. For sufficiently small & 6: h these quantities are of the Same magnitude as or smaller than those terrns which are 1 O(hq+l) in the Bsense. In the following the terminologies " O(hq+ l ) for sutheiently small r = "

1x1 or " O(hq+') for E -t 0 " are always to be understood in this sense. Such terms are of Course not

(I O(hq+l) in the rigorous sense: if h tends to 0 and X is kept iixed, i.e. a fixed problem is considered, then there are always values of h such that & < h is violated (even for Re(X) << O), and so it is not justified to speak of O(hq+')-terms; but for stepsizes of practical relevance (adjusted to the smoothness of the Solution being sought and guaranteeing a certain desired level of accuracy),

<< h is often realistic. I Al

Let ue now continue the discussion of the terms (3.7). For the mildly stiff case ( lXlh w 1 ) the I terms (3.7) cannot be considered to be O(hq+') because << h is not true. For increasing values of U, however, the O(hq+l)-level is again achieved by and by. (Note that for U > 1 the term $ of (3.7a) is not present - as simple calculations show - and that instead of the (Xh)aeXh-terms of (3.7) 1 terms of the form (M,)'2'~ appear, which tend to 0 for U -+ m.)

I Summarizing, we observe that (2.20) is satisfied in the above sense at all grid points if the situation is strongly stiff. Thus (2.19) shows Ru = O(hq+') at all grid points. If the situation is mildly stiff, there is an order reduction within the C, and consequently within the Ru at the first grid points.

I But in the scalar case it can easily be shown that these order reduction effects of the Ru at the

1 first grid points are damped away with increasing U by higher and higher powers of -. However, it is important to notice that for non-scalar problems the damping effect just described

I cannot be concluded from estimates based on (2.18): In the vector case there will usually be non-stiff

'fQ+2-i *) Note that I,,( depends on =e1(t) = O(IX(q).

eigenvalues of moderate size besides the stiff eigenvalue X. Since the one-sided Lipschitz constant rn is influenced by these non-stiff eigenvalues (note that m 2 max Re(Xi), X; . . . eigenvalues of f )

Y : the (&)"-factors (cf (2.18)) have no darnping power: in many cases m > 0 and then (&) increme8 with U; but also if m < 0 and Iml is of moderate size, hlml is small and (in contrast

to (&)", lXlh PJ 1 ) the quantity decreases very slowly with V, such that a t a fixed t = tu E [O,T] , Ru has not the desired order O(hq+'). Nevertheless, it will turn out in part 2 of our paper that similar darnping properties hold as in the scaiar case. Our analysis will be based on completely other techniques than B-convergence estimates.

in our simple example the typical effects are easily overlooked; but it is a non-trivial task to show that asymptotic error expansions analogously exist for a large class of stiff problerns. In Section 4 we will present a singular perturbation analysis for the strongly stiff case.

4. GENERAL THEORY

We will now discuss the question whether the structure of the solution of the variational equations (V.E.'s) observed in our scalar model(3.1) extends to more general Situations. In our model positive powers of X appeared in the ek(t), but only in conjunction with exponentially decaying solution components. The question arises whether it is possible for general systerns (2.1) that smooth solution components of the ek(t), which do not exponentiaily decay, are also affected with positive powers of X (where the parameter X is a quantity of magnitude of the stiff eigenvalues). Notice that a term of the form hkXk-2(Xt,)j2ty (appearing in (3.5)) can reduce to h2 in the worst case (i.e. in the mildly stiff case a t the first grid points). On the contrary, " hkX' (smooth function (tu )) " is unbounded as X + -00. Intuition tells that the occurrence of such a term cannot be compatible with the B-convergence properties of the implicit Euler scheme which hold uniformly w.r.t. A. To confirm this intuition we will apply singular perturbation techniques. We shall see that the Situation in the general case is quite analogous as for the model from Section 3.

In the general case (2.1) the V.E.'s are of the form

where g(t) depends in a recursive way on certain derivatives of the solutions of the preceding V.E.'s. In the foiiowing we assume that z(t) is a smooth solution of (2.1) in the Spirit of the B-theory. Let

We assume that the complex-valued quantities cl (t), c2 (t), T(t) and T" (t) are smooth functions (i.e the derivatives which appear in the following are assumed to exist and to be of moderate size). Furthermore, we assume that Re(cz (t)) > 0. s > 0 is a small real parameter. According to (4.2a,b), fy (t, z(t)) = O(&"). W.r.t. the higher derivatives of f we assume that they are smooth, i.e.

Our analysis will refer to the ;?-dimensional case (i.e. c t (t) and c2 (t) are scalar functions). However, all considerations can also be understood in the sense that the C, (t) are vector-valued functions. The more general case of several clusters of stiff eigenvalues would require a more general, multi- parameter ( E ~ , E Z , . . . , E ~ ) theory which would cause no fundamental complications.

The question arises whether our smoothness assumptions can be relaxed. To our opinion - based on numerical experience - this is not possible. If, for instance, higher f-derivatives are large (i.e. (4 .2~) is violated) then probably the structure of the global error will be totally destroyed. Only the O(h)-B-convergence bound remains valid.

To enable the application of singular perturbation techniques we transform the V.E.'s (4.1), using the matrix T(t) which diagonalizes fy (t, z(t)) (cf. (4.2a7b)). With

(4.3) Z(t) = T-' (t)e(t) ,

(4.1) transforms into

Using the denotation

(4.5) ( t ) = ( ) , A(t) = 1 a2,1(t) t ) a1,2(t)) a2,2(t) ' g(t) = (QI:!) and multiplying the second component of (4.4a) with E , we get

This is a special (linear) case of an equation of the type

which is usually considered in the singular perturbation theory (cf. e.g. 07Malley [9]). Notice that the singular perturbation theory applies to (4.6) because we have assumed that Re(cl(t)) > 0.

We have supposed that the ci(t) and ~ , ~ ( t ) do not depend on the parameter E. On the other hand, for many models with a stiff eigenvalue X (where E is identified with - i) smooth solutions z(t) depend on X (i.e. on E) in a harmless way, i.e. on non-negative powers of E. Consequently, fy (t, z(t)) depends on e too. Therefore, in (4.6) we would have to work with expansions C, (t) = ci,o (t) + ECi,l (t) + . . ., A(t) = & (t) + &Al (t) + . . .. We have conveniently restricted ourselves to E-

independent data since this causes no substantial differentes in our singular perturbation analysis. All the material presented below could easily be rewritten for the slightly more general case of edependent data functions. The really essential thing is the E -dependen~~ of the ek(t) (and not of z(t)) which involves negative powers of e in the inhomogenous terms of the subsequent V.E.'s due to the differentiation with respect to t.

1

4.1 Singular Perturbation Theory for the First Variational Equation

Let us now consider the first V.E. where g ( t ) = $T-' ( t ) r t t ( t ) does not depend on c (due to our remark above) . Introducing the " stretched variable"

C we rnake the following singular perturbation ansatz for Z(t) = el ( t ) :

~ ( t ) = X o ( t ) + & x l ( t ) f ~ ~ ~ 2 ( t ) - k & ~ x 3 ( t , ~ ) f E ~ - ~ ( T ) + E ~ ~ ~ ( T ) + E ~ ~ ~ ( T , & ) ,

b (4-9) y(t) = YO ( t ) + cY1 ( t ) + c2 ~2 ( t ) + e3 ~3 ( t , 8 ) + n- 1 ( T ) + eno ( T ) + c2 nl ( T ) + c3 n2 ( T , E ) .

J

The purpose of our ankatz up to c3 is to demonstrate the basic ideas of the singular theory. Expansions upito an higher E-power will be used for a full discussion of the solutions of the V.E.'s and their influerice on the C, and b, (cf. (2.9), (2.12)).

b Up to now we have not fixed the starting values z(O), y(0) for the equations (4.6). Let

(4.10) can be rnotivated as follows: Recall that [O,Tj is a subinterval of the whole integration B interval with constant stepsize; the starting value e l (0) is defined by the value of the function il ( t ) a t the endpoint t = 0 of the preceding subinterval (cf (2.17)). In the first of these intervals the

B starting value of el is 0. Our structural analysis will show that (4.10) is always justified if we look a t the recursive concatenation of all of these subintervals (cf. Subsection 4.2).

Inserting the ansatz (4.9) into (4.6) and using Taylor expansions for the data functions in the terms B involving n7.i ( T ) and q ( T ) , we obtain *)

z'(t) = X; ( t ) + E X ; ( t ) + c2 X i ( t ) + e3 X i (t , E)+

S! (g1.P) J0 UO!?n10s aq? snq& -0 = 3 aq qsnw ?uewuo:, uo1?el8a3u! aq1 JOJ a:,!oq:, Jno -a.! 'CO +- 1 SE olaz 09 pua? (I)~U'(J)~ leq? a~rnbal aM Ä~oaq? uo!?eq~n?lad ~e1n8u!s aq? U! pnsn 6.e 8u:paa~o~d E (0) '-U pue 0 E (0) 1-U ~eq? (qg1.v) UIOJJ SMOIIOJ 91 .O = (~)O/I - 0 = (0)1-U amq aM (qzl-P) 03 Cdu!p~o:,3e 'a~ow~aq~m~ -q?oours an (eg~'~) U! suor?:,unj oqap aq? a:,u!s q?oours s! (?)ox .(az~.~) 09 8u!pJ0331? oz = (0)Ox anp Bu!J~e?s aq? q?!~ uo!?enba Te!?uaJaj!p Jeau!] qe3s e JO uo!?nlos aq? SI (?)OX snq~, '0 r (7) O/I saqdur! (egl-p)

- [(3 '1) zu '3 + (J) 1U z3 + (J) OU3 + (J) -U] -

+ [(3 'J) 1~~3 + (J) OUl z3 + (2) 1 -W31 -

+ [(3 'J) zu C3 + (1) Ku z3 + (J) ou3 + (J) -U]

1 Coefficients of E' :

(4.15~) X: (4 = (CI (4 + U 1 , l (t))Xl (t) + a1,2 (t)Yl (t),

0 = - ~2 (t) Y1 (t) + u2,l (t) XO (t) +so-+ ~ ( t ) -&I# .+

1 Yl(t) is uniquely determined by the second equation in (4.15a). Thus XI (t) is iixed together with the initial condition Xi (0) = z i - m- 1 (0) = zi (cf. (4.12a)). Furthermore, the starting value (0)

1 is fixed via (4.12b) : no (0) = yl - Yl (0) = yl - - e-l(0)zo c2 (0) - CZ(O) ' It follows frorn (4.15b) that

(Again we have k e d C = 0 according to lim (T) = 0.) 7- W

For higher powers E' the procedure continues in the following way: K(t) is the solution of a purely algebraic equation which is recursively defined by X;,l (t) and X- 1 (t). Once Y , (t) is known, X, (t) is given by the solution of a scalar linear differential equation. The starting value is fixed by (4.12a): X; (0) = - mi-2 (0). Furthermore, ni-l(O) = yi - Y;. (0) by (4.12b). ni- 1 (r) can be determined as the solution of a simple linear differential equation with constant coefficients. m;-l (T) *) is obtained by a pure quadrature with integration constant C = 0. It can be Seen that (r) and

1 (T) are always of the form

B (4.17)

with certain polynomials p(r). The polynornial coefficients can explicitly be expressed by (0), C: (0) , . . . , G, j (0) ,U:, (0) , . . . and r(0) , s(0). Note that (4.17) is bounded independently of E

on [0, $1 and is rapidly decaying away from 0.

What remains to be done is to estirnate the remainder terms of our E-expansion. In the case of (4.9) it foliows from (4.11) that they must satisfy equations of the foliowing form:

1 Coefficients of :

X.& (tt E) = (cl (t) + U 1 , l (t))X3 (t, E) + a1,2 (t)Y3 ( t t E), (4.18'3) EY3/(t, 8) = -C2 (t)Y3 (ty 8) E[a2,1(t)X3 (t, 8) + a2,2 (t)Y3 (t, E)]+

+ u2,1(t)Xz(t) + a2,2(t)Y2(t) - Yl(t)Y

*) A special case occurs when equating coefficients of E' immediately before the E'+'-remainder terrns (in particular for i = 2 in the the special case of (4.9)): In contrqt to r~+~( r ) , the quantity mi - i ( r ,~ ) is not discussed for the rnoment but is considered as a part of the remainder term. Therefore it is discussed together with the E=+'-terms (in varticular with i + 1 = 3 for our ansatz (4.9); cf. (4.18b)).

SI a?eurllsa Bu!?~nsa~ aql '(3 JO s~amod q9:q A~luayxps q?!~ (1) Vtu( tl)O ,3 UJOJ aq? 30 surla? ~apu:eura~-aolAq, 30 pue (LI.%) adAl aq3 JO SUO!~U~J JO SUO~~BU~~WO~ ~eau!~ azo ~fi puo @ ar>u!s A~a?e!paurur~ SMOIIOJ q3~q~)

dq paqeurysa aq usr> ZU '(3 ';)~ul pue (3 '1)s~ '(3'2) EX 'uaq~ 4

:Id'oj lenla?u! leu?a!~o aq? uo 1 alqe!wa aurq aq? -?.J.M s,d~~ o?u! suoysnba Jno a183SaJ o? InJaSn s! q! pua s!q? o,~, '([g] ~aqnq~aqn 'p!auq~s 'qmq a3ue?sur ~oj .33) ul queqsuo3 ~?!q3sd!~ pap!s-auo aq? uo paseq saqeur!?sa ~?MOJ~ ü~ouy-~~a~ uroq ABM p~epueqs Q U! pau!eqqo aq ue3 (8T.P) U! ma?sAs aq? 30 (3':) zu '(3';) Iw pue (3'1) EA '(3 'J) EX suolqnIos aqq ioj spunoa

Furthermore, estimates for the derivatives (w.r.t. t) of the remainder terms can be derived by dif- ferentiation of the defining equations (4.18a7b) and recursive use of bounds for the lower derivatives. This yields

Analogous bounds hold for Y3, m l and n2.

Summarizing, we See that el (t) = T(t)El (t) splits up into two terms:

(4.26a) Ei (t) := T(t) ( X0 (t) + &Xl (t) + . . . .Yl (t) + . . .

is called the " outer solutionn , as usual in the singular perturbation theory.

(4.266) & (f) := T(t) ( c2mo(C) + . . . ~ n o ( f ) + E 2 n l ( f ) + ...

is the rapidly varying and exponentially decaying solution component (the so-called "inner solu- tion" ) .

4.2 Extension to Further Variational Equations

Let us now consider the V.E7s for all ek(t), k = 1,. . . ,q, and their recursive dependence. We have recursions on three levels:

- level 1: recursive definition of the ek(t), k = 1,. . . , q, - level 2: concatenation of several subintervals . . . , [0, T] , . . . for ek (t), - level 3: singular perturbation expansion of ek(t) on [O,T].

The inhomogenous terms of subsequent V.E.'s contain derivatives of ei (t). (4.26b) and Sm(!) =

E-* $,(T) show that these terms will be affected with negative powers of E, i.e. terms of the form

will arise. (In the general, nonlinear case there are also terms of the form ~ - j ~ ( t ) e - ~ ? ( ~ ) ~ f gener- ated by terms as for instance f,.,., (t, z(t))e:.)

In view of this Situation the central question is:

If the right-hand side of a V.E. contains negative powers of E only in conjzlnction uith stretched variable inhomogenous terms, is then the same true for the solution of the respective V.E.? I.e., are there solution components affected vith E-i-factors in the inner solution only ?

In accordance with our intuitive argument from the beginning of Section 4, it will turn out that this is indeed the case: carrying out the recursion at level 1, we will show that E-'-factors appear only with stretched variable terms in the solution as well as in the inhomogeneities of the V.E.'s.

Furthermore, this will justify our ansatz (4.10) for the starting values in each subinterval with constant stepsize (recursion at level2) for all V.E.'s: At the endpoint t = T of the actual integration

interval [O,T1 the inner solution components (even with negative powers of E) are irrelevant due to their rapid exponential decay. Therefore the ansatz for the starting values on the next subinterval does not contain negative powers of E (for the definition of these starting values See (2.17)).

T h e central question raised above is not trivial as the following simple example shows: The IVP

has the solution

with a critical term O(e-l)et in its first component. This shows that in general outer solution terms affected with negative powers of e occur also in cases where in the inhomogeneity only stretched variable terms are affected with negative powers of e. So it is by no means trivial to show that such effects cannot arise for the V.E.'s.

We will now discuss the technical details of all these questions. First of all, note that we have used a short ansatz (4.9) up to ~~-1eve l to make the ideas behind the singular perturbation techniques transparent. For our purpose we must now extend the ansatz up to a considerably higher eL-level. We then have analogously to (4.24) and (4.25):

# L E' -XL (t, E) = O(E'-'-~), e -X (t, E) = o(eL-'-I),

(4.28) dtt dt

cl6mL-2 = 0(c1-'-l), , & dti

e - n ~ - ~ (:,e) = ~ ( e ~ - ~ - ~ ) . dtt

In the inhomogenous term of the i-th V.E., S e l (t) appears, and for 1 > i negative powers of E

can be avoided in (4.28). This is important since the differential equations defining the quantities Xi(t, E) , X(t, C) , m[-2 (C ,E ) , n[-l($,e) are of a much more complicated structure - they are of the Same type as the V.E.'s themselves - than the equations defining the terms Xj(t), Y,(t), ( j < I ) , mj($) , ( j < 1 - 2) , n j ( i ) , ( j < 1 - 1). For the quantities Xl(t,e), K(t, E), ml-2($,e), nL-l(;,e), only estimates in the sense of (4.28) can be derived and negative powers of E must be avoided. For the preceding terms Xj(t), Y, (t), m j ( t ) , n j ( i ) the Situation is totally different: in particular, the quantities m j ( t ) , n j (i) are solutions of linear differential equations with constant coefficients and can therefore be deteimined explicitly such that the effects of negative powers of E (appearing in the inner solution) can easily be discussed.

In a similar way as for el (t) the ansatz for the ek(t), k < i , is chosen such that negative powers of E are not present in the respective derivatives of the remainder terms.

We will now discuss the " centrnl question" raised above in context of the singular perturbation theory. 1.e. we have to discuss whether all outer solution terms affected with E-'-factors vanish. Let us assume recursively (recursion at level 1) that for ej(t), j < i, this is the case *):

* ) For j = 1, (4.29a) is not correct; in this case we must write the outer solution terms as E-j+1X~,j(t)+~-j+2~l,j(t)+. . . = eoXO,l(t)+elXl,l(t)+. . . (cf. (4.9)) and the second component analogously. The inner solution terms, however, are correct in (4.29a) also for j = 1, since m-l(r) = n-l (T) 0 for the first V.E. (cf. (4.14)).

&-j+2 X o, j ( t ) + €-i+3Xl,j(t) + - - -+ ~ - ' X j - ~ , j ( t ) + ~ O X j - ~ , j ( t ) + . . .+ + ~ - j + ~ m o , , ( f ) + ~ - j + ~ m ~ , , ( f ) + . . .

( 4 . 2 9 ~ ) ej ( t ) = T ( t ) &-'+'& Cl ( t ) + ~ - j + ~ ~ , j ( t ) + . . . + ~ ~ i ; - ~ , , ( t ) j ( t ) + . - . +

+,-j+Z % , j ( : ) + ~ - j + ~ n ~ , ~ ( $ ) + . . .

with -

Xo,j(t) E Xl , j ( t ) E.. - = Xj-3, j ( t ) 0 , (4.29b) Yo,j(t) G Yl,j(t) . . - 5 - 3 , j ( t ) G Y3'2,j(t) 3 0.

L

t Each of the quantities r n k j ( 2 ) and nk , , ( f ) is of the form p( j)e-c2(0)i.

d-i+l The inhomogeneity of the i-th V.E. contains -ej(t), j < i7 and therefore (due to (4.29a)) the

9 inhomogeneity of the transformed V.E. reads )

C E - ~ + ~ u o , ; ( ~ ) + E-'+~uI,; (t) + . . . + e-I U;-3,, ( t ) + . E ~ U ; - ~ , , ( t) + . . - + +,-i+z uo,;( f ) + &-i+3ul , ; (~) + . . .

E - ' + ~ V O , ~ ( t) + ~-~+~vl,i(t) + - . - + E O K - ~ , ~ ( t ) + E ' K - ~ , ~ (t) + . . . + +&-i+Z voli (:) + C - ~ + ~ V ~ , , ( ; ) + . . .

Ci Again, the quantities uk,; ( 4 ) and uk,; ( f ) are of the form p( 5)e-c2(0)3.

in order to complete the recursion we have to show that (4.29a) and (4.29b) yield 3

with

(4.31b) Xo,;(t) = Xl, i ( t ) = . . . = Xi-3,,(t) = 0>

Yo,; (t) Y1,i ( t) = . . . X-3,;(t) X-z,;(t) = 0.

Analogously to (4.13a), (4.15a) the equations defining the outer solution components of Z; ( t) are:

*) The inner solution terms of e,(t) are (cf. (4.29a)). Therefore the corresponding

stretched variable terms of &i+i due to T- ' ( t ) . (e,(t)) = T- l ( t ) . &i+l ,j-i+ 1

dti-j+i (T( t )e j ( t ) ) = ej(t) + . . ..

g< W;.

[where for k = 0 we set X1 , i ( t ) - Y-l,,(t) V-i,i(t) 0).

(4.33) Xk,;(O) = 0 for k = O(l)i - 3,

we can conclude recursively (using (4.30b) and (4.32))

Xk,,(t) 3 0, k = O(1)i - 3, (4.34)

YkVi (t) I 0, k = O ( 1 ) i - 2,

and are able to establish the desired assertion (4.31b). The gap to be closed in our argumentation is to verify the validity of (4.33). Due to the starting condition

(4.33) is equivalent to

Recall that mk,, (T) is always of the form p (~ )e -~2 (O)' (since the integration constant is always set to Zero to satisfy lim mk,; (T) = 0). Thus (4.36a) means that the constant coeficient of the respective

'+W

polynomial AT) must vanish.

For a number of V.E.'s we have carried out the recursions, i.e. we have explicitly determined the respective mk,, (T), nkli (T). It turned out that, indeed, the constant coefficients of the polynomials within the inner solution terrns mk,i(r), nk,,(r) vanish for negative powers of E. Therefore the de- s i r d property - that no outer solution components with negative powers of E occur - is satisfied in accordance with our conjecture at the beginning of Section 4. An algebraic proof of the validity of (4.36a) will be presented in Subsection 4.4. Instead of this algebraic proof (which is rather techni- qual) it wo,uld probably be possible to give an indirect proof for (4.36a), based on a contradiction to the B-convergence properties of the implicit Euler scheme (cf. the discussion at the beginning of Section 4). We did not elaborate such a proof due to the following reason: In Subsection 4.3 we will discuss the main question of part 1 of our Paper, namely which properties of the e;(t) ensure the full asymptotic expansion for e -, 0. It will turn out that the existente of such an expansion can be concluded if instead of (4.36a) the sharper condition

is satisfied, i.e. the constant coefficients of the polynomials within the inner solution terrns mk,i(r), nk,i(r) vanish not only for negative powers of e but also for E". As mentioned above, the validity of mkli(0) = 0, k = O(1) i - 4, could probably be concluded indirectly; the relations nk,;(0) = 0, k = O(1)i - 2, follow simply from the respective relations Yk,;(t) G 0, k = O(1)i - 2; but the most difficult problem is the verification of mk,, (0) = 0 for k = i - 3. The latter relation could only be proved algebraically. This proof Covers also the cases k = O(1) i - 4 - and so we did not try to give other arguments for (4.36a).

4.3 A Property which Ensures a Full Asymptotic Expansion for e -t 0 t2

Recalling that the RV1s satisfy the difference equation (2.12), we find that, according to (2.19), the desired result

(4.37) Ru = O(hqf ') for sufficiently small e

holds if b, - C, = O(hq+') holds for e -t 0 a t all grid points tu (V = 1,2,. . .). (Concerning the terminology "O(hqf ') for sufficiently small e " cf. the discussion at the end of Section 3.)

First of all we introduce the denotation (cf. (4:26a,b), (4.31a,b)) *)

(4.38~) E k (t) := T(t) eOXk-2,k(t) f elXk-l,k(t) f '-) 9 k=2 ,3 , ... , Q ' Y ~ - ~ , ~ ( ~ ) + . . .

Thus,

The hk-' I,,k-terms in (cf. (2.9))

can now be written as

hk-1 IUlk =hk-1 . hq-k+2 i e(q-k+2) - ( q - 1

(t, - uh) (1 - u ) ~ - ~ + ' & - 0

(4.40) - - hq+' 1 $- k+2) (9 - k + I)!

(t, - oh)(l - u ) ~ - ~ + ' & + 0

h a l l that b, is defined as the collection of all terms involving higher f-derivatives at O(hq+ ')-level. Consider for instance t he case q = 2 (cf (2.7b)) :

1 b~ = - ~ U U (tu , z(ty)) . [2h3el (t,)e2 (t,) + h4 e; (t,)] + 2

(4.41) I

+ l / f u Y Y ( t Y , ~ ( t u ) 2 +u(hel(t,) + h2e2(t.)))(l - u12& [hel(t,) + h2s( tU) l3 0

*) For k = 1 the smooth variable terms cannot be written as in (4.38a); cf. the footnote in conjunction with (4.29a); see (4.26a) for the definition of EI (t).

18

(The 0(h2)-term 4 fyy ( tu,z( tu))h2e~(tu) in (2.7b) appeared already in the second V.E. (2.11).) TO ( split b, into smooth- and stretched variable terms, we insert (4.38~) into (4.41). For the first term i

we obtain 1 1 2

fyy (tu, %(tu)) . [2h3 (EI (tu ) + SI (-))(.G e (tu) + ~2 (-)) e + h4 (G (tu) + s2 (L)) ] = (4AZa) &

4 2 =, fyy (tu, %(tu)) . [2h3 (Ei (tu)& (tu) + h E2 (tu)] + -stretched variable terms;

each term containing at least one factor & ( C ) is of Course of "stretched variable typen, e.g. f fYY (tu, %(tu.))- [2h3 (Ei (tu)& (Y) ] . Analogously, the integral remainder term in (4.41) spiits into

+stretched variable terms.

By dehition, the stretched variable term in (4.42b) is the difference of the "full" integral remainder term in (4.41) and the "smoothn integral remainder term in (4.42b). By the Mean Value Theorem it can easily be Seen that this difference is indeed of stretched-variable type because it is of magnitude 0(hs1( C ) + h2 S 2 (C)).

The extension of the Splitting (4.42) to the general case q > 2 is straightfon&ard.

I According to (4.40) and (4.42) the inhomgeneity of (2.12) can be wntten as

(4.43) bu - C, = Ku + Lu,

( where Ku and L, denote the collection of all smooth- and stretched-variable terms, respectively. For the smooth-variable terms,

(4.44) K, = 0(h4+l)

is tfivial; tbe critical point is to show

8' where - due to the structure,of Sk(:) - the most critical case is Y = 1. Instead of discussing (4.45) directly (which would be simple but somewhat lengthy), we prefer another more short and transparent argumentation. To this end we derive an equivalent formulation of (2.12): With the abbreviation

(4.46) V, (t) := hei (t) + . . . + hq e, (t),

the asymptotic expansion (1.1) reads

(4.47) C, =z( tv)+vq( tv) + Ru.

Inserting into the implicit Euler scheme (2.2), using (2.1), expanding f(t„(,) and rearranging yields the foiiowing differente equation for Ru :

1 1 -(R" - &-I) = . f ( ~ , ) R, + ~'(t.) - j-[z(tv) - ~ ( t , - ~ ) l + h

1 (4.48)

+ f y (tw.(t,))v,(tu) - j-[v,(t.) - ~,(t,-l)]+

where j ( ~ , ) is deiined as in (2.7a). (4.48) is of Course equivalent to (2.12)~ but the inhornogeneity 1 is written in another way. Again we split the inhornogeneity into smooth- and stretched variable i i terms (analogously as in (4.42)), ending up with

d

K: and Li denote the smooth- and stretched- variable terms, respectively. Since 11 f, (tu, z(+)j 11 is large, fy (tu, z(t.)) (hEl (tu) + . . . + hq E, (t,)) seems not to be a smooth term; but using the V.E.'s we can rewrite fy (t,, z(t,))vq(t,) as

+..., and then there is no problern to split up into srnooth- and stretched variable terms. (Recall that the higher f-derivatives fy fy yy , . . . are assumed to be srnooth, cf. (4.2~) .)

Obviously,

(4.51) K, +L; = K; +L;

holds; moreover,

(4.52) K, = Ki , L, = L;,

which can easily be shown by the following argurnent: Let

1 -(Sv - Sv-1) = f(tY>~V)> (4.53) h

So = ~ ( 0 ) + hEl(0) + . . . + hqEq(0).

The asymptotic expansion for reads

(4.54) & = ~ ( t . ) + h 2 ~ ( t , ) + ...+ hq2,(tU)+ R,,

where the first V.E.

1

(4.55) ?(t) = f y (t,~(t))El(t) + p"(t)>

e1 (0) = E1 (0) I

has the srnooth solution Zl (t) z El (t) , because the outer solution El (t) is itself a solution of the first V.E. . Similarly, the second V.E.

1 1 1 4( t ) = fy (t, z(t))E2(t) - -zm(t) + 5Ey(t) + Z> fyy (t, z(t))E: (t) =

6 (4.56) 1 1 1 = f y ('1 ~(t))El(t) - --'W) + :Ei(t) + 5fYY (tl z(t))E? (t), 6

A

has the smooth solution e2(t) E2 (t), and so on. Ru satisfies a difference equation

or equivalently

where the inhomogeneities are defined in exactly the Same way as in (2.12) or (4.48) but depend on the ek (t) Ek (t) instead of the ek (t). Thus,

and the inhomogeneity in (4.58) is equal to K:. Again the inhomogeneities in (4.57) and (4.58) coincide, and therefore Ku r K:. This and (4.51) imply L, E L:.

So it is now sufiicient to discuss (instead of (4.45) ) the validity of

(4.60) L: = O(hq+l) for e-+ 0.

tu- 1 For u > 1 it can be Seen from (4.48) that all terms of L: contain a factor e-cz(0). or e - C 2 ( 0 ) T (or are at least of magnitude O(e-Cz(0)%)). Thus all these terms rapidly vanish for E -+ 0 and can be considered O(hqf ') for sufficiently small E. Only for v = 1 (where e-c2(0)- = eO = 1) there remains one critical inhomogenous term, originating from - [-V, (tu-

Sl (0) + hS2 (0) + . . . + hq-lSq (0) = 0(hq+l) is obviously satisfied for e -+ 0 if (4.36b) holds, since then all nonvanishing terms in (4.61) are affected with positive powers of e. Thus, (4.36b) is indeed a sufficient condition for the existence of a full asymptotic expansion at all grid points in the strongly stiff case (cf. the discussion at the end of Subsection 4.2).

In the following we will say that the " e0-property " is satisfied if the constant coeffi- cients of the polynomials in the inner solution terms of the ek(t) vanish for negative powers of e und jor eO.

4 mentioned at the end of Subsection 4.2, there is no intuitive Bconvergence argument for the eO- property. Indeed, the ~ ~ - ~ r o ~ e r t ~ does not hold in general for Bconvergeat methods, e.g. for the B- convergent implicit midpoint rule. Hotuever, & kplicit Euler acheme haa the e0-property.This will be proved in the next Subsection, concluding our s -+ 0-analysis for asymptotic error expansions. +. .

4.4 Algebraic Proof of the eO-property

i The generd form of the V.E.'s (where the first and second are given in (2.10) and (2.11)) reads

+ terms witA higher f-derivatives.

The transformed V.E.'s (cf. (4.4)) are

6 (t) = A(~)z; (t) + A(~)z, (t) + (-l)"' T-l(t)z(i+l) (t) + 2 WT-l(t)(T1-k+l)(*) (t)+ (4.63)

(i + I)! k= 2

k !

1 1 + ZT-l(t)fyy ( t y z ( t ) ) ( ( ~ ~ l ) ( t ) , (Tc;-l)(t)) + iT-'(t)fYy(t, z(t))((TZ2)(t), (Te;-~)(t)) + . . .. The terms within the sum in (4.63) read

(-l)k~-l(t)(~G-k+l)(k)(t) k! =

-- - k! (i) T"(~)c!*;:)~ (t) + . . .]

(4.64) - -- ( - I ) ~ - ~ e(k-l) (-11 k-2

~ ( ~ ) ~ ( k - 2 ) k! A(t)-a-k+l(t) + (k - 2)! a-k+l(t) f " '

1 with A(t) := -T-'(t)T'(t), Ä(t) := -T-'(t)Tf'(t) (cf .(4.46)).

2

To keep the presentation transparent we will prove the eo-property only for i 5 4. (The extension to i > 4 is straightforward but requires a considerable technical effort.) For this purpose it is necessary to investigate the leading inner Solution terms up to e0 (cf. (4.31a)):

&Om,2(r)Y

: (4.65) 3'd V.E. !- * "O mo,3 (T), .- >, %4 * \; < < +=

,&&&F> <:

&-l%,3(cf), @n1,3(~),

.4tk V.E. "-1%,4(r)1 &Om1,4(r)? ; ,

&F ri . e-2120,4(r)? e1n1 ,4 ( r ) ~ ' ~ 2 , 4 ( ~ ) r

1 .

'; lt has to be shown that these terrns vanish for r = 0. 1 simply follow from the initial condition

$ ** :- 0; the only nontrivial case is i = 4, 1 = 2, where Y2,s

&'. Wequence of w,4 (0) = 0 (see the discussion at the he verification of the ~ ~ - ~ r o ~ e r t y for the first comp

For the second compon

, (t) G 0 is not a priori obv end of this Subsection).

lonents mr,;(r).

ious The

For i = 4 we have therefore to look at the leading terms mt~,~(r) , no,4(r) and m1,4(~), nl14(r) (note 1 that nl,;(r) must be known to express ml,,(r)). F'rom the recursive definition of the V.E.'s it can

be seen that this requires the investigation of the respective leading terms mt~, , (~) ,no , , (~) and m1,; (T), n1,, (T) fo all i 5 4 (and not only those appearing in (4.65): The inhomogeneity of the 4th

1 % (transformed) V.E. contains, for instance, the term 24 s E l (t) (cf. (4.64)) ; due to

% it can be seen that &mo,l (T), (T) appear in the inhomogeneities of the equations for -

d= motr (T), n0,4(r) *) (together with $m,2 (T), d ; r ~ , 2 (T), . . .). Analogously, the equations for 6 6 22-

m1,4(7), n l ,4(~) depend On d;rmi,i (T), dZrnt,l (T), dz3 mo,l (T), . . . (cf. (4.79) below).

Let us now discuss the influence of the higher f-derivatives f„ (tu ,z(tu)), . . . on the equations for W,, , nr,j, 1 = 0,l. The V.E.'s contain respective inhomogenous terms of the form:

I ld V.E. -

1 + ,fYYYY 0 7 z(t))e:(t)

The Q (t) split into smooth terms E, (t) and stretched-variable terms Si (9. For the equations =. defining the inner Solution those terms in (4.68) are relevant which contain at least one factor s;(:) (cf. the discussion in Subsection 4.3). Equating coefficients of powers of s in the Singular perturbation context (where the second components of the equations are multiplied by E), it turns out that

i) -,,(T), not; (T) are not influenced by the terms in (4.68);

ii) the inhomogeneity of the equation for ml,, (T) contains a term of the form bt,2 ( o ) X ~ , ~ (O)t~g,;-~ (T) originating from

(4.69) T-l(t)fyY(t,z(t)) (Tel(t),TZ;-~(t)) =: B(t)(El,Ei-~(t))

at t = 0 (after Taylor expansion). nl,;(r) is not influenced by the terms in (4.68).

(The detailed verification of i) and ii) above is simple and left to the reader.) In the sequel we will abbreviate the above coefficient by

(4.70) b(0) :=b;,2(0)&,l(O).

Our proof of the ~ O - ~ r o ~ e r t y is based on recursive representations for the rnl,,(T) and n1,; (T) which we derive below. Differentiation w.r.t. T will be denoted by

(4.71) 1 := ( T ) , k = 0>1> 2,. . . We wiU use the definitions

(4.722) mi,;(~):=O, n , , ( ~ ) : for i < 0 ,

* ) Recall that, before equating coefficients, the second component of the V.E.'s is multiplied by E.

23

I (4.73b) &,i(r) := nr,i(r) + ~t(O)n~,i- l (r) .

i a1.2(0) Note that Ar,; (T) = mit (T) + ;;onr,i (T).

1 I = 0 : Rmrsion mn,i. nn,i

I m,, = %,,(T) and no,; = noli(r) satisfy (cf. (4.63) and (4.64))

j i

'-I J-1)' [k] (4.740) mt! = al,2 (~ )%, i f C w m t i - k + l + a1,2 (0) C k! nO,i-k>

k=2 k= 1

With no,o = 0, (4.74) can be rearranged into

(-I)& [kj (4.750) - ai,2 (O)m,i = C k! mO,,-k+i

' (-1jk [k] + a1,2 (0) C b! n~, ; -k+~- l ' k= 1 k=l

(4.75b) y ields ( m , is cancelled): The linear combination (4.75a) +

This implies

and thus

for all j >: 1. Therefore, rjlo,j(r) G C. Our requirement that all inner Solution terms tend to Zero for r -, oo yields C = 0. It follows that

pua I-?'b aqq 103 sp~oq qq~) (951-P) uo~an3ai aqq Bupn pua (81'~) rep ''? 69 "OtU Supsaidx3

((~9-V) pure ($9'~) -33) A~qqes (J) = f'Iu pm (J) "Iur = ?'Iw

t LU*: LW rod uo.zssn3ag : T = 1

&,i analogously, as can easily be seen), we can rewrite (4.81) in the form

9

k= 1

C1 2 (0 ) [k ] (4.82) + - b(O) [kl

C2 ( 0 ) n~,i-k+i-i h(0)"o1i-k+1-1 - I -

k= 1 C2 ( 0 )

+ "i, ( ~ ) n t ! - k - ~ - I Now let j 3 1 be fixed. For all i = 1,. . . , j - 1, (4.82) is [ j - i - 11-times differentiated. For i = j (4.82) is integrated (with integration constant C = 0 as usuai for inner solution terms), which is indicated by the superscript [ - I ] . This results in the following recursion for the 6i[ri1, i = 1,. . . , j:

We will now evaiuate (4.83) for r = 0. To this end it is necessary to treat those terms in (4.83) which are affected with T-dependent coefficients in a special way. E.g., consider

i (4.84) $$(rnt/-k+l)[j-i-l~ (0) .

k=l

First of ail, the relation

(4.85) [rn(r)]['I (0 ) = 1 n['-'] (0 ) , 1 = - l , O , 1,2,. . . , y ields

(,Ik] )[j-i-11 [j-i+k-21 - 0,i-k+l (0 ) = ( j - i - 1) noli-k+l (0 ) -

(4.86) = ( j - i + k - ' 1 n~,i-k+l [j-i+k-21 ('1 - %,i-k+l [j-i+k-21 ( O ) - - =(rno,i- k+l) [j-i+k-1](0) - k n[j2+k-2] 0,s k+i (O)'

Inserting (4.86) into (4.84) we obtain

( - 1 ) [ k ] [ . '- C ~ ( ~ ~ O , i - k + l ) J - $ l I(0) = (4.87) k= 1

(-1Ik i [j-i+k-1] ( 1 ) Ij-i+k-21 = C -k](rno, i -k+l) ('1 - E yk n~,i-k+i ('1-

k= 1 k= 1

26

1

Duc to (4.75b) arid %,o = 0, the second sum in (4.87) can be expressed as

(4.87) and (4.88) yield the desired reformulation of (4.84).

The second T-dependent term in (4.83), namely

is treated analogously as (4.84).

S-arizing ail that, the evaiuation of (4.83) for r = 0 results in the triangdar system of j linear

i equat ions

for the unknowns a12(0) i i I 1 alj-4 + = ii>!;'1(0) -.-a1,1(0)4,~~, (0) + U:,, (0) [(rno,i-i)ij-il (0) + ~ ; i ~ j n ~ , , - ~

C2 (0)

(Here we have recombined most of the %,,-terms from (4.83) into io,,-terms [cf (4.73b)I.)

The desired representation for ni1,,(0) is now obtained as follows: The linear system (4.90) has the unique solution s, = 0, i = l(1) j. For i = j the [ j - i]-th derivatives in (4.91) are 0-th derivatives. s, = 0 yields

a1,2 (0) M

fii,j(0) = mi,j(O) + c2 (0) nljj(0) =

aO-vrovertu @r j = 1.2: 3: 4 :

For e,(t) the so-property is equivalent to

-1aded mo jo q~ed U! papaau Xl?p!~dxa aq 11!fi (26'~) pue (~L-P) '~aaafio~ -X?~adold-~i aq? JO JOOJ~ aq? JOJ X~essa3au qou -M (26'~) pue (8t.P) suoyelal aql U! squapgao:, aq? jo arnpnlqs pal!eqap aqL .yrourar lou!~

'0 = (0) P'OUr UIO~ SU!MOI~OJ la??e1 aq? '0 (3) V'IX pue 0 E (3) P'IA JO a3uanbasuo3 e s! xas?! qqa 0 (3) p'z~ asne~aq 0 = (0) *'zu alouIlaq?q

-0 = ((0) Z'Ou'(0) &'ou'(~)*'ou'(~) &"uL(o) v'lu) 40 uoyvurqtuo3 rvauy = (o)*'h.u (LCP)

?Qq? sfioqs (26'~)

-0 s (3) "'"X PUB 0 E (3) P'% JO aauanbasuo3 e s! Ras?! q~!qm 0 E (J) P'IA asneaaq 0 = (0) ""U azouiraqlqg

-0 = (0) &*OU(O) ZIItl -(o)'4<-- =(o)*'Oui (96-p)

sfioqs (8~-P) -0 E (J)*'% pue uo!?!puo:, @!?!U! aqq 03 anp 0 = (0)"o~- = (0)"Ou aAeq afi u!esv

*((qrl-P) '53) 0 = (3) E'Ox

PUQ 0 E (I)&'% JO abuanbasuo:, Q s! Jrasq qqfi 0 E (~)&'IA asneaaq 0 = (0)s'Iu aroulaq?m,g

SfiOqS (8L.P) -0 (3)&'0!4 pue no!?!puo:, le!?!u! aq? 0% anp 0 = (O)E'OA- = (o)&'ou aAeq afi upsv

:W

'(2'~ uo!$3asqns U! sanpa Su!r)~e?s aq? jo uo!ssn3qp aq? P)

REFERENCES

[I] G.Bader arid P-Deuflhard, A semi-implicit mie for stif systerns of ordinary dif- fmential equations, Num. Math., 41 (1983), pp. 373-398.

[2] J.C.Butcher, A stability ptoperty of implicit ~unge-Kutta rnethods, BIT, 15 (1975), PP. 358-361.

[3] G.Dahlquist, Error analysis fot a Class of rnethods for stifnonlinear Mitial vdue problems, in: Lecture Notes in Mathematics 506, G.A. Watson (Ed.), Springer-Verlag, Berlin, 1976.

[4] G.Dahlquist and B.Lindberg, On sorne irnplicit one-step rnethods for stiff differential equa- tions, Dept. of Computer Sciences, Royal Institute of Technology, Rep. TRITA-NA-7302, 1973.

[5] K.Dekker and J.G.Verwer, Stability of Runge-Kutta rnethods for stiff nonlinear differential equations, North-Holland Publ., Amsterdam-New York-Oxford, 1984.

[6] R.Frank, J-Schneid and überhuber, er, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981), pp. 753-780.

[7] R.Frank, J.Schneid and überhuber, er, Stability Properties of miplicit Runge-Kutta Meth- ods, SIAM, J. Numer. Anal., 22 (1985), pp. 497-515.

[8] R.Frank, J-Schneid and überhuber, er, Order results for irnplieit Runge-Kutta MethodP applied to stiff systerns, SLAM J. Numer. Anal., 22 (1985), pp. 515-534.

[9] R.E.07Malley Jr., Introduction to singular perturbations, Academic Press, New York - Lon- don - Toronto - Sydney - San Francisco, 1974.

[10] H.J.Stetter, Analysis of discretitation rnethods for ordinary differential equations, Springer- Verlag, Berlin, Heidelberg, New York, 1973.

[LI] M-Veldhuizen, Asymptotic ezpansions of the global error for the irnplicit rnidpoint rule (stiff cczse), Computing, 33 (1984), pp. 185-192.