Geometric proofs of numerical stability for delay equations

Geometric proofs of numerical stability for delay equations Nicola Guglielmiy

Dip. di Matematica Pura e Applicata, Universita dell'Aquila, via Vetoio (Coppito), I-67010 L'Aquila, Italy, e-mail: [email protected] and Ernst Hairer

Dept. de Mathematiques, Universite de Geneve, CH-1211 Geneve 24, Switzerland, e-mail: [email protected] In this paper, asymptotic stability properties of implicit Runge Kutta methods for delay dierential equations are considered with respect to the test equation y (t) = a y (t) + b y (t 1) with a; b 2 C . In particular, we prove that symmetric methods and all methods of even order cannot be unconditionally stable with respect to the considered test equation, while many of them are stable on problems where a 2 R and b 2 C . Furthermore, we prove that Radau IIA methods are stable for the subclass of equations where a =  + i with ; 2 R, suciently small, and b 2 C . Subject Classi cations: AMS(MOS): 65L20; CR: G1.7. 0

1. Introduction

In this work we study the asymptotic stability behaviour of numerical methods when applied to the test equation

y0 (t) = a y(t) + b y(t 1); t > 0; (1.1) with initial condition y(t) = g(t); 1  t  0, where the constants a; b and the function g(t) are all complex. We are mainly interested in methods that produce unconditionally stable numerical solutions for all a and b, for which the exact solution

tends to zero. Most of the publications on this subject restrict their analysis to the cone jbj < 0. We use

bm(y) = ei m ('(y) y) b? (my);

(4.2)

which follows immediately from (3.1) and (2.3). We shall show that the whole root locus curve bm (y) (with the exception of the point b = a) lies outside of Q? [a]. This is obvious for my  , because jbm (y)j2 = jb? (my)j2 = (my)2 + a2 is monotonically increasing, and the maximal distance of Q? [a] from the origin is jb? (a )j with a 2 (0; ) given by a = a cot a (Fig. 1). For 0 < my < , this follows from 0 < '(y) < y (Lemma 4.1) and from (4.2), because the factor

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N. Guglielmi and E. Hairer

ei m ('(y) y) represents a clock-wise rotation with an angle of less than my degrees. Since the intersection of Q?[a] with the circle fb ; jbj = jb? (my)jg is a connected arc containing the left intersection point with the real axis, this implies that bm (y) lies outside Q?[a] also in this case. The statement of the theorem now follows from the fact that all b 2 Q? [a] \ R lie in Qm [a] \ R (Theorem 2 of Guglielmi & Hairer

2

(1999)).

Remark 4.3 An interesting consequence of the previous result is that Gauss

methods are stable when a is real and in particular, according to the de nition given by Barwell (1975), they are Q-stable.

4.2 Non-symmetric methods We next consider arbitrary non-symmetric stability functions, and we give sucient conditions for numerical stability. Similar to the proof of the previous theorem, we study conditions, under which the numerical root locus curve does not enter the analytical stability region. For a xed a  1, the point bm(t) of (2.7) is certainly outside of Q? [a], if its distance to the origin is larger than b? (a ) with a as in the proof of Theorem 4.2. Because of jbm(t)j = jmz (t) aj, this is the case if mz (t) 62 D[a], where 



D[a] = z 2 C ; jz aj < jb? (a )j : The set D[a] for several values of a, and the union D=

[

a1

(4.3)

D [ a]

(4.4)

are shown in the left picture of Fig. 4. Computing the envelope of this family of circles, one nds that the set D lies to the left of the curve given by the imaginary axis for jyj  , by 2  ; x() =  sinsin cos 

p

2 2  y() =   sin  sin cos 

for  2 (0; )

p

and by the circle with radius 1 and center 1 between the points (3=2;  3=2). Theorem 4.4 Let a be real. In the situation of Lemma 2.2, assume that Qm [a] \ Q? [a] 6= ; for all m, and that for every t at least one of the following two conditions is satis ed: (i) x(t)  0 and y(t)  '(t), (ii) z (t) = x(t) + iy(t) 62 D. Then, it holds Qm [a]  Q? [a] for all m  1. Proof. Because of Qm [a] \ Q? [a] 6= ;, it is sucient to prove that the numerical root locus curve bm (t) never enters the analytical stability region Q? [a]. If mz (t) 62 D, which is the case for jmy(t)j   or when condition (ii) is satis ed, this is true by

9

Geometric proofs of numerical stability 4 12 2 8 −4

−2

2

4 4

−2

−4

4

8

12

Fig. 4. The left picture shows the discs D[a] of (4.3) and their union D. The picture to the right shows the stablity domains of the Radau IIA methods with s = 2; 3; 4; 5 stages together with the boundary of D. By Table 1 of Guglielmi & Hairer (1999), condition (i) of Theorem 4.4 is satis ed for all values between the origin and the point marked by an asterix.

de nition of D. We therefore have to consider only the case where x(t)  0 and 0  m'(t)  my(t) < . The idea is to write the root locus curve (2.7) as   bm (t) = ei m '(t) e i m y(t) b? (m y(t)) + m x(t) ;

(4.5)

and to interpret this equation geometrically as follows (Fig. 5): we start by considering the point b? (my(t)) on the boundary of the analytical stability domain Q? [a], we multiply it by e imy(t) (this is a rotation around the origin and yields the point imy(t) a on the vertical line 0: (5.3) Moreover, assuming that Qm [a0 ]  Q? [a0 ] for a real a0 < 1, and that the boundary of Qm[a] touches Q? [a] only at the point b = a, then the condition (5.3) implies that Qm [a]  Q?[a] for all a in a complex neighbourhood of a0 . Proof. The rst statement follows from bm (') b? (m') = ( 1)k mC'2k + O('2k+1 ); and from the fact that close to b? (0) = a the region Q?[a] lies to the left of the curve b? (). The second statement follows from a continuity argument. 2 We remark that (5.3) is also a necessary condition for  (0)-stability (see Guglielmi & Hairer (1999)). The stability function of the Radau IIA methods satis es (5.3), so that the stability result of Theorem 5.2 applies.

6. Conclusions

In this work we have concluded the characterization for A-stable symmetric methods when applied to the test equation (1.1). In particular, this class of methods which has been proved to be stable for real coecient delay dierential equations (under some further technical assumptions, as shown by Guglielmi & Hairer (1999)) performs stably on equation (1.1) when a is real and b complex, but turns out to be unstable on general complex coecient problems. Moreover, we have shown by a local perturbation analysis that no method of even order can be stable for the general complex coecient test equation (1.1). On the other hand we have shown that Radau IIA methods (with a number of stages s  7) perform stably on problems with real a, and also in the case where the imaginary part of a is suciently small. A complete proof of  -stability for the Radau IIA methods is still missing.

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N. Guglielmi and E. Hairer References

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