A Global Convergence Theorem for a Class of Parallel Continuous ...

A Global Convergence Theorem for a Class of Parallel Continuous Explicit Runge-Kutta Methods and Vanishing Lag Delay Dierential Equations C.T.H. Baker & C.A.H.Paul Numerical Analysis Report No. 229 (revised) May 1994

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A Global Convergence Theorem for a Class of Parallel Continuous Explicit Runge-Kutta Methods and Vanishing Lag Delay Dierential Equations Christopher T.H. Baker and Christopher A.H. Paul May 23, 1994 Abstract

Iterated continuous extensions (ICEs) are continuous explicit Runge-Kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay dierential equations (DDEs). ICEs have a particular r^ole in the explicit solution of DDEs with vanishing lags. They may be regarded as parallel continuous explicit Runge-Kutta (PCERK) methods, as they allow advantage to be taken of parallel architectures. ICEs can also be related to a collocation method. The purpose of this paper is to provide a theorem giving the global order of convergence for variable-step implementations of ICEs applied to state-dependent DDEs with and without vanishing lags. Implications of the theory for the implementation of this class of methods are discussed and demonstrated. The results establish that our approach allows the construction of PCERK methods whose order of convergence is restricted only by the continuity of the solution.

Key words. Parallel continuous explicit Runge-Kutta methods, iterated continuous extensions, delay dierential equations, vanishing lag. AMS subject classi cations. 65L06 65L70 65Q05 65Y05

1 Introduction The outline of this paper is as follows: We state the equations de ning a class of methods [1] for solving delay dierential equations (DDEs). Our main aim, achieved in Section 3, is to establish the global convergence and order of convergence for our class of methods. Section 2 provides preliminary results to this end. In Section 4 a variant method is introduced, and we show how the analysis of Section 3 can be modi ed to analyse this method. Practical considerations are addressed and numerical results are provided in subsequent sections. The presentation covers variable-step, varying smoothness and the possibility of a vanishing statedependent lag. The paper provides a systematic, self-contained and rigorous analysis valid under these conditions. We assume familiarity with the concept of a continuous explicit Runge-Kutta (CERK) method (see [5] p.176 et seq.) for initial-value problems in ordinary dierential equations (ODEs), y0 (t) = f(t; y(t)) for t  t0 and y(t0 ) = y0 :

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The CERK methods outlined in this paper are suitable for solving DDEs of the form u0(t) = F(t; u(t); u( (t; u(t)))) for t0  t  tN  T;

 Mathematics Department, Victoria University of Manchester, Manchester M13 9PL, England.

1

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with (t; u(t))  t and u(t) = (t) for t  t0 , where (t) is a prescribed initial function. In particular, these methods are suited to DDEs which have a singular or vanishing lag, for which (t; u(t)) ! t as t ! t . Other work relating to the explicit numerical solution of vanishing lag DDEs is referenced by Neves [8]. It is a complicating feature of (2) that the solution u(t) is liable (depending on the initial function (t)) to suer jump discontinuities in its derivatives at an ordered set of points ftj g; however, on successive intervals [tj ; tj +1] the solution is smooth. A recent paper [11] describes a strategy for calculating the points ftj g for a system of DDEs; see also [4]. A CERK method can be associated with the tableau

c

A

 bT() ; where A is strictly lower triangular, and with the points

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ftn; ftnigg with tn+1 = tn + Hn and tni = tn + ci Hn (Hn > 0): For our adaptation of a -stage CERK method to the DDE (2), we shall employ the parameters

c = [ci]i=1 with c1 = 0, ci 2 [0; 1], A = [aij]i;j=1, b() = [bi()]i=1 (a vector of polynomials in ) and B = [bj (ci )]i;j=1. (The matrix B corresponds to a collocation method with abscissae fcig.) We use

these parameters in an iterative method for computing an approximate solution ue(t), for which a result of the form supt 0 has continuous order p := p(c; A; b()) when used to solve the initial-value problem (1), if the approximation ye(t) satis es supt2[t0 ;t0+H ] jye(t) ? y(t)j = O(H p +1 ) as H & 0 whenever y(t) and f(t; y) have suciently smooth derivatives. The continuous order is governed by conditions on the CERK triple (see [1], p.374). We shall assume that the derivative function F(t; u1; u2) and the solution u(t) of (2) are such that the following holds:

Hypothesis 1.3 There exists an integer p such that the CERK method applied to the initial-value problem y0 (t) = f(t; y(t)) for t 2 [tn ; tn+1]; with

y(tn ) = u(tn) and f(t; y(t)) = F(t; y(t); u( (t; u(t)))); yields an approximation ye(t) that satis es supt2[tn ;tn+1 ] jye(t) ? y(t)j = O(Hnp+1 ) as Hn & 0 uniformly for each n = 0; 1; . . .; N ? 1.

The value of p is speci c to the problem (2), with the given choice of (t). If u(t), (t; u) and F(t; u1; u2) are all suciently smooth then p  p (in general p = p ). Henceforth we assume the following: 2

Hypothesis 1.4 The following additional assumptions hold: 1. F(t; u1; u2) has continuous partial derivatives upto order p , 2. u(r?1)(t) 2 Lip[tn ; tn+1] for all n = 0; 1; . . .; N ? 1, and 3. u(s?1)(t) 2 Lip[tn; tn+2] for all n = 0; 1; . . .; N ? 2.

The iterative methods discussed here are de ned by iterated continuous extensions (`ICEs'), which were introduced into the open literature in [1]. The practical motivation for such methods was illustrated by Paul & Baker in [10]. ICEs are de ned as follows:

De nition 1.5 In the numerical solution of (2), employing the CERK triple fc; A; b()g, an initial approximation zen[0] (t) to ue(t) for t > tn de nes: (i) starting values fue[0] (tni )gi=1 satisfying ue[0] (tni) = ue(tn ) + Hn

i?1 X j =1

[0] )); aij F(tnj ; ue[0](tnj ); zen[0] (e nj

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and (ii) for m  1, an m-th ICE ue[m] (tn + Hn ), on setting k = m in the recurrence:

ue[k] (tn + Hn ) = ue(tn ) + Hn ze[l] (t)

where n

=

8 > e(t) < u zen[0] (t) > : [l] ue (t)

 X i=1

[k?1] )); bi()F(tni ; ue[k?1](tni ); zen[k?1](e ni

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t  tn ; t > tn and l = 0; t > tn and l  1:

if if if

[k] = (t ; ze[k] (t )). Here ue(t) = ue[m] (t) if t0  t  tn , ue(t) = (t) if t < t0 and eni ni n ni

According to this de nition, ue[k] (t) (for k  1 and t 2 [tn; tn+1]) is a continuous extension based on the [k?1] ))gg. Note also that ze[k] (t)  u values fue(tn ); fF(tni; ue[k?1](tni ); ezn[k?1](e ni e[k] (t) for t  tn and k  1. n [0] For each interval [tn ; tn+1], given an initial approximation zen (t), the m-th ICE is uniquely de ned by the choice of CERK triple and \step" Hn. Two possibleP initial approximations are: Case 1 ue(tn ), and Case 2   ue(tn?1 +  Hn?1) for   1 { that is, ue(tn?1) + Hn?1 i=1 bi ( )F(tn?1;i; ue[m?1] (tn?1;i); zen[m??1 1] (e n[m??1;i1] )) extrapolated from the preceding interval [tn?1; tn]. In the latter case, when n = 0, there is no \preceding interval"; we then recommend the choice ze0[0] (t) = ue(t0 ). Remark: The m-th ICE may be regarded as an approximate solution to the equation ue(tn + Hn) = ue(tn) + Hn

 X i=1

bi ()F(tni ; ue(tni); (tni ; ue(tni ))):

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2 Assumptions and lemmas The aim of this paper is to obtain a result for the m-th ICE of the form sup ju(t) ? ue(t)j = O(H  ): t 0 and n = 1 + Cnhn, where 0  Cn  C , for n = 0; 1; . . .; N ? 1 and PN ?1 h  T . If fDn g and fen g are non-negative sequences with i i=0 en  nen?1 + Dn hn for n = 0; 1; . . .; N ? 1

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and e?1  0, then

en  (e?1 + 0max fD gT) exp(CT): kn k Proof. Denote by k;n the quantity

n Q j =k

j . From the recurrence (7), by induction,

en  0;ne?1 + 1;nD0 h0 + 2;nD1 h1 +


+ Dn hn (n  0): P

Note that n  1 + Chn  exp(Chn ). Thus k;n  exp(C nj=k hj )  exp(CT) for k = 0; 1; . . .; n and n = 0; 1; . . .; N ? 1. Also D0 h0 + D1 h1 +


+ Dn hn  max fDk gT for n = 0; 1; . . .; N ? 1, and the result k follows. We also require the following assumption:

Hypothesis 2.2 The sequence of steps fHng is assumed chosen such that u(r?1) (t) 2 Lip[tn ; tn+1] (in particular, u(r) (t) 2 C[tn; tn+1]) and Hn+1=Hn = n   for n = 0; 1; . . .; N ? 1. The derivative function F(t; u1; u2) in (2) satis es uniform Lipschitz conditions (with Lipschitz constants L2 and L3) with respect

to its second and third arguments. In the case of a state-dependent lag, we also require that (t; u) satis es a uniform Lipschitz condition (with Lipschitz constant L ) with respect to u, and the solution u(t) satis es a uniform Lipschitz condition (with Lipschitz constant Lu). It is further assumed that is evaluated to ensure, for every computed approximation zen[l] (t) to u(t) with t 2 [tn; tn+1], that (t; zen[l] (t))  tn+1 .

Notation: In the theorems that follow we shall refer to the Lipschitz constants L2, L3, L and Lu , and the

constants K1 = L3 + L3 LuL and K2 = L2 + L3 + L3Lu L . b;b b ()g indicated The m-th ICE is related to the CERK method de ned by the CERK triple fbc; A below:

c

b

Ab

 bb T()



c A 0





.. . B 0 .. . 0 ... ... .. .

.. .

... ... ...

0 .. . .. . .. .

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c 0


0 B 0  0T





0T bT() ;

where dim(bc) = m dim(c). This tableau allows standard Runge-Kutta (RK) analysis to be applied to the discrete RK method [1] obtained by setting  = 1. The discrete RK method has a Butcher tableau corresponding to a class of \parallel RK methods" [6]; ICEs, when applied to either an ODE or a DDE, can be implemented eciently on a parallel computer (see Section 5.3). The continuous order conditions determining p (see De nition 1.2) for the tableau (8) applied to the ODE (1) (and similarly for the DDE (2) when (tni ; u(tni))  tn for all i) were discussed by Paul & Baker [10].

Hypothesis 2.3 The parameters fc; b()g de ne the continuous quadrature order q := q (c; b()) of the CERK triple, for which the continuous extension based on the values fu(tn); fu0(tni)gg satis es: (i) max u(t + Hn ) 01 n

? u(tn ) + Hn

 X

4

i=1

! bi()u0 (tni )

 C Hnminfq ;rg+1

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when u(r?1) (t) 2 Lip[tn; tn+1], and (ii) max u(t + Hn ) 11+n n

? u(tn ) + Hn

when u(s?1)(t) 2 Lip[tn ; tn+2].

 X i=1

! bi()u0 (tni)

 Cn Hnminfq ;sg+1

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P

The CERK triple is said to provide a continuous quadrature extension u(tn) + Hn i=1 bi()u0 (tni) to u(t). Using standard Peano-type quadrature theory, the CERK triple fc; A; b()g satis es (9) and (10) when  j +1 X bi()cji = j + 1 for j = 0; . . .; q ? 1: (11) i=1

Note that these conditions are independent of n. b;b b ()g (8) cannot exceed q , and if m Remarks: The continuous order p of the CERK triple fbc; A is suciently large it equals q . In the preceding hypothesis, we have introduced the generally distinct integers r and s  r related to the dierentiability of u(t). However, a more precise insight can be obtained by introducing integers pn, rn and sn such that u(rn ?1)(t) 2 Lip[tn; tn+1 ] and u(sn ?1)(t) 2 Lip[tn; tn+2]. In fact, r = min0iN ?1frig and s = min0iN ?2fsi g. The determination of pn (corresponding to p in Hypothesis 1.3) requires information on the behaviour of the lag, since (t; u(t)) can traverse several previous intervals as t traverses the interval [tn ; tn+1]. Lemma 2.4 (Dierenced F-values). Assuming that Hypothesis 2.2 is valid,

jF(t; v; ezn[l] ( (t; zen[l] (t)))) ? F(t; w; u( (t; u(t))))j  L2jv ? wj + K1 sup jzen[l] (x) ? u(x)j xtn+1

for t 2 [tn; tn+1].

Proof. Writing (t; zen[l] (t)) as z , we use the triangle inequality

jF(t; v; ezn[l] ( z )) ? F(t; w; u( (t; u(t))))j  jF(t; v; ezn[l] ( z )) ? F(t; v; u( z ))j + jF(t; v; u( z )) ? F(t; w; u( z))j + jF(t; w; u( z )) ? F(t; w; u( (t; u(t))))j; and exploit the restriction (by hypothesis) that (x; zen[l] (x))  tn+1 for x 2 [tn ; tn+1]. In particular jF(t; v; ezn[l] ( z )) ? F(t; v; u( z ))j  L3jzen[l] ( z ) ? u( z )j  L3jzen[l] (x ) ? u(x )j for some x  tn+1 ; and

jF(t; w; u( z )) ? F(t; w; u( (t; u(t))))j  L3ju( z ) ? u( (t; u(t)))j  L3LuL jzen[l] (t) ? u(t)j: Notation: For n  0, let n[k] := sup ju(t) ? zen[k] (t)j and en := sup ju(t) ? ue(t)j = maxfen?1; n[m] g: t0ttn+1

t2[tn ;tn+1 ]

Lemma 2.5 (Bounds on n[0] ). Case 1. If zen[0] (t) := ue(tn), then n[0]  en?1 + Lu Hn: Case 2. If zen[0] (t) := ue(tn?1 +  Hn?1) for n  1, then fq ;sg+1 n[0]  en?2 + M Hn?1n[m??1 1] + C Hnmin ?1 for positive constants C and M . 5

Proof. Case 1. From the de nition of zen[0] (t),

n[0] =

 

sup ju(t) ? ue(tn )j t2[tn ;tn+1 ] en?1 + sup ju(t) ? u(tn)j t2[tn ;tn+1 ] en?1 + LuHn:

Case 2. Suppose that u(s?1)(t) 2 Lip[tn?1; tn+1], and the continuous quadrature order of the CERK

triple fc; A; b()g is q . We introduce the continuous extension U(tn?1 +  Hn?1) based on the values fu(tn?1); fu0(tn?1;i)gg, namely U(tn?1 +  Hn?1) = u(tn?1) + Hn?1

Thus, by Hypothesis 2.3,

 X i=1

bi ( )u0(tn?1;i):

fq ;sg+1; u(tn + Hn )  U(tn?1 +  Hn?1) + C Hnmin ?1 where  = 1 + Hn=Hn?1 2 [1; 1 + ]. From the de nition of zen[0] (t) and applying Lemma 2.4, fq ;sg+1 n[0]   sup ju(tn?1 +  Hn?1) ? ue(tn?1 +  Hn?1)j + C Hnmin ?1 

 2[1;1+]

 en?2 + Hn?1  sup  2[1;1+]



 X

fq ;sg+1 jbi ( )jK2n[m??1 1] + C Hnmin ?1 

i=1 [ m ? 1] fq ;sg+1 ; en?2 + M Hn?1n?1 + C Hnmin ?1  P

where M = K2  sup jbi( )j.  2[1;1+] i=1 Notation: Let  X N = sup jbi()j 1 ? L jjKA2 jj H : 2 1 n 01 i=1

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Lemma 2.6 (A bound on n[1] ). Given the CERK triple fc; A; b()g and the value p in Hypothesis 1.3, there exists a positive constant U such that for L2jjAjj1Hn < 1, [1] p+1 n[1]  N Hnn[0] + d[1] (13) n where dn  (1 + N Hn)en?1 + U Hn : Proof. Consider the internal-stage derivatives

f [0] ni

= F(tni; ue(tn ) + Hn

and the related quantities fni = F(tni; u(tn) + Hn Using Lemma 2.4 and K2 = L2 + K1, Now

i?1 X j =1 i?1 X j =1

[0] ; ze[0] ( (t ; ze[0] (t )))); aij fnj ni n ni n

aij fnj ; u( (tni ; u(tni)))):

[0] [0] ? f j  K2en?1 + K1n : max j f ni ni i 1 ? L2jjAjj1Hn

zen[1] (tn + Hn)  ue[1] (tn + Hn) = ue(tn) + Hn u(tn + Hn) = u(tn) + Hn 6

 X i=1

 X i=1

bi ()fni[0] ; bi ()fni + O(Hnp+1 );

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by Hypothesis 1.3. Dierencing these equations, employing (14), and bounding the order term using a suitable constant U , we nd that n[1]  en?1 + Hn sup

 N Hn

 X

01 i=1 [0]  + (1 + N H

n

+ K1n[0] + U H p+1 jbi()j K1 ?2 enL?1jjA n jj H 2

n)en?1 + U Hnp+1 ;

1 n

since K1  K2 . The proof of a similar result for n[k] with k  2 follows dierent lines:

Lemma 2.7 (A bound on n[k] for k  2). Assuming that Hypothesis 2.3 is valid and that L2jjAjj1Hn < 1, then

n[k]  N Hnn[k?1] + d[nk] where d[nk]  (1 + N Hn)en?1 + C Hnminfq ;rg+1 for k  2:

Proof. We have by Hypothesis 2.3 that

u(tn + Hn)  u(tn) + Hn and, from the de nition of the k-th ICE, ue[k] (tn + Hn) = ue(tn) + Hn

 X

 bi ()F(tni ; u(tni); u( (tni ; u(tni)))) + C Hnminfq ;rg+1

i=1  X i=1

[k?1] )): bi ()F(tni ; ue[k?1](tni); zen[k?1](e ni

Thus, dierencing these equations and using the triangle inequality, n[k]  en?1 + Hn sup

 X

01 i=1

jbi()jj
fni[k?1] j + C Hnminfq ;rg+1 ;

where for k  2, using Lemma 2.4 and K2 = L2 + K1,

j
fni[k?1] j = jF(tni; u(tni); u( (tni; u(tni)))) ? F(tni; ue[k?1](tni ); zen[k?1](e ni[k?1] ))j  L2 n[k?1] + K1(en?1 + n[k?1]): Therefore, since K1  K2, where Q = K2 sup

 P

01 i=1

 n[k]  QHnn[k?1] + (1 + QHn)en?1 + C Hnmin fq ;rg+1 ;

jbi()j = (1 ? L2 jjAjj1Hn)N < N .

3 The convergence result We now present the main result, concerning the order of convergence of ICEs when applied to vanishing lag state-dependent DDEs. We assume that the CERK triple fc; A; b()g is given, and let H := maxn fHng. We suppose that Hypothesis 2.2 is valid (so that u(r?1) (t) 2 Lip[tn; tn+1 ] for all n) and assume that the conditions of Hypotheses 1.3, 1.4 and 2.3, in terms of q and r, hold for the CERK triple and the problem (2). If ue(t) is computed from the m-th ICE using (8), then the following results hold:

Theorem 3.1 (Global order of convergence).

Case 1. If zen[0] (t) := ue(tn ) for each interval [tn; tn+1], then

sup ju(t) ? ue(t)j = O(H 1 ) where 1 = minfm; q ; rg:

t0t