Katholieke Universiteit Leuven Department of Computer Science

Transcription

1 Regarding the absolute stability of Störer-Cowell ethods Syvert P. Nørsett Andreas Ashei Report TW 601, October 2011 Katholieke Universiteit Leuven Departent of Coputer Science Celestijnenlaan 200A B-3001 Heverlee (Belgiu)

2 Regarding the absolute stability of Störer-Cowell ethods Syvert P. Nørsett Andreas Ashei Report TW 601, October 2011 Departent of Coputer Science, K.U.Leuven Abstract Störer-Cowell ethods, a popular class of ethods for coputations in celestial echanics, is known to exhibit orbital instabilities when the order of the ethods exceed two. Analysing the absolute stability of Störer-Cowell ethods close to zero we present a characterization of these instabilities for ethods of all orders. Keywords : ultistep ethods for second order probles, Störer-Cowell ethods, absolute stability. MSC : Priary : 65L06, Secondary : 65L20, 70M20.

3 Regarding the absolute stability of Störer-Cowell ethods Andreas Ashei, Syvert P. Nørsett Departeent Coputerwetenschappen, K.U. Leuven, Belgiu Institutt for Mateatiske Fag, NTNU, Trondhei, Norway Eail: October 3, 2011 Abstract Störer-Cowell ethods, a popular class of ethods for coputations in celestial echanics, is known to exhibit orbital instabilities when the order of the ethods exceed two. Analysing the absolute stability of Störer-Cowell ethods close to zero we present a characterization of these instabilities for ethods of all orders. 1 Introduction Many echanical probles in physics are expressed as second order differential equations of the for, y = f(x, y), y(x 0 ) = y 0, y (x 0 ) = y 1,0. (1) Nuerical algoriths for such equations are arguably as old as echanics itself: Isaac Newton was the inventor of what we today call the Störer- Verlet schee[5], one of the ost popular schees for echanical probles. Carl Störer, eponyous to the Störer-Verlet ethod, published his ethods aied at coputing orbits of charged particles in Earth s agnetic field(aurora borealis) as early as in Störer s ethods is a class of ultistep ethods containing the Verlet ethod as a siplest case. A related class of ethods is Cowell s ethods, dating back to These two classes of ethods, Störer s and Cowell s or predictor-corrector pair of these, are still very popular in nuerical astronoy. Störer ethods of order 13 and 14 aied at high precision calculations appear in recent research publications like [11, 4], and are ipleented in several progra packages like the popular NBI-package fro the UCLA astronoy group[15]. Granted, these ethods are generally not energy or syplecticity preserving, and they thus fall outside the ore recent field of geoetric nuerical integration wherein we find ethods that are thought to be ore appropriate for echanical probles. There are however still good reasons to consider non-geoetric integrators. In particular if the ai of the coputation is very high precision where the right qualitative behaviour coes Supported by the Norwegian Research Council s Espedahl Stipend 1

4 iplicitly with the high precision. In recent works involving high precision coputations have traditional ethods like e.g. Störer s ethods been central (see for exaple investigations on the stability of the outer solar syste [7] and [1]). High order ethods with tiny step sizes are used for such coputations. However, stability and round-off probles for sall step sizes ust be carefully analysed and treated in in order for these high precision coputations to be truly high precision[15]. The ethods are known as prone to rounding errors probles when running with sall step sizes. This is described, aong other places, in Wanner, Hairer & Nørsett s book[6], where a stabilisation of the schee is also proposed. In this work we shall address the question of stability of Störer-Cowell ethods for sall step sizes. A basic result here is Dahlquist s classical result fro 1976 which bars unconditionally stable ethods of order exceeding two[2]. The ethods are easily proved to be zero-stable. However, when considering the absolute stability, as defined by Labert[9] we shall see that they are not necessarily stable at any point close to zero. This is strongly related to a well known property of Störer-Cowell ethods, naely that they suffer fro what is called orbital instabilities when the order of the ethod exceeds two; whenever integrating a circular orbit the nuerical solution will either spiral inwards or outwards. This has been the background for the developent of syetric ultistep ethods[10, 13] and ethods that integrate trigonoetric polynoials exactly[3]. Here we will go back to a ore thorough analysis of the stability of the Störer- Cowell ethods and characterize exactly which ethods are absolutely stable and unstable for sall step sizes. We shall also see exaples ethods that are unstable for sall step sizes, but exhibit stable regions away fro zero. Stabilisation schees will not be discussed in this work. Let us end this introduction with a few words about the history and tiing of the this work. This paper is naely based on a aster thesis by Even Thorbergsen fro 1976[14] and an unfinished note by Nørsett fro the year after. The note was however put in a drawer and forgotten. Recently, while cleaning his office in preparation for his retireent Nørsett found the note, and in discussions with his forer PhD student Ashei it was concluded that the note could, in a reworked and finished for, be of interest to the nuerical analysis and celestial echanics counity. 2 Multistep ethods for second order probles We start by reviewing soe relevant facts regarding ultistep ethods for second order probles. Full expositions of the following theory are given in books by Henrici[8], Labert[9] and Wanner, Hairer & Nørsett[6]. We are considering second order initial value probles of the for (1). Multistep ethods for such probles are of the general for, α j y n+j = h 2 Defining the linear difference operator L[y(x); h] = β j f n+j. (2) [ αj y(x jh) h 2 β j y (x + jh) ], 2

5 we say, using the notation of Henrici[8], that the ethod has order p if its expansion in h around zero is of the for L[y(x); h] = C p+2 h p+2 y (p+2) (x) + C p+3 h p+3 y (p+3) (x) +..., (3) for any sufficiently sooth y(x). We say that the ethod is consistent if it has order at least one. Defining the generating polynoials, ρ(ζ) = this is equivalent with, α j ζ j, σ(ζ) = β ζ j, (4) ρ(1) = ρ (1) = 0, ρ (1) = 2σ(1). (5) Central to this work is two concepts of stability, both which are stated by Labert[9]. Definition 1. We say that the ethod (2) is zero stable if all roots of ρ(ζ) are contained in the unit disk, and those roots that are located on the unit circle are of ultiplicity at ost two. Definition 2. The ethod (2) is called absolutely stable for a given q C if and only if all the roots of the stability polynoial ϕ(ζ) := ρ(ζ) + q 2 σ(ζ), are contained in the unit disc. Otherwise we call the ethod absolutely unstable for this q. 2.1 Störer-Cowell ethods We shall consider ultistep ethods of the for y n+k 2y n+k 1 + y n+k 2 = h 2 β j f n+j. (6) These ethods are usually referred to as Störer-Cowell ethods. One way to arrive at such ethods arise when adding the Taylor series for y(x n + h) and y(x n h), y n+1 2y n + y n 1 = h 2 y (x, y n ) + h4 12 y(4) (x, y n ) + h6 360 y(6) (x, y n ) +... Störer s ethods are obtained by in this case replacing derivatives of y(x, y n ) with backward differences. Truncating and eliinating higher order ters generates ethods of arbitrary order. We define the generating polynoials for these ethods, ρ(ζ) = ζ k 2ζ k 1 + ζ k 2, σ(ζ) = Adapting the notation fro [8] we denote by y n+1 2y n + y n 1 = h 2 3 β j ζ j. (7) σ j j f n, (8)

6 j σ j σ j j σ j σ j Table 1: Coefficients for Störer s (σ) and Cowell s (σ ) ethods. the k-step Störer ethod of order k + 1. is the backwards difference operator: z n = z n z n 1. Likewise we have the k-step iplicit Cowellethod of order k + 1, y n+1 2y n + y n 1 = h 2 σj j f n+1. (9) Translating into ordinate for (6) is achieved by applying the forula ( ) f j = ( 1) l f j l. l l=0 Coefficients of these ethods are obtained in a ore straightforward way than the above described expansion by interpolation on the right hand side f(x, y), for Störer s ethods yielding 1 σ = ( 1) (1 s) 0 1 [( s ) + and for Cowell s ethods, 0 [( ) s σ = ( 1) ( s) + ( )] s ds, (10) ( s + 2 )] ds. (11) Nuerical values of coefficients are ost conveniently coputed by recursion forulas[8]. Table 1 gives values up to k = Soe properties of Störer-Cowell ethods For the following stability analysis we will need soe results regarding the ethods. The following proposition essentially states that the coefficients of Störer s ethods are positive and decreasing, negative and increasing for Cowell s ethods. Proposition 1. The coefficients of Störer s ethod (10) satisfy with equality only for = 1, and σ 0, 1, σ +1 σ, 2, with equality only for = 2. The coefficients of Cowell s ethod (11) satisfy σ +1 0, 3, 4

7 with equality only for = 3, and with equality only for = 4. σ +1 σ, 4, Proof. We shall only do the proof for the case of Störer s ethods. The proof for Cowell s ethod is copletely analogous. Since this proof is based on anipulation of binoial coefficients we will refer to the following identities[12], ( ) ( ) t t t + 1 =, (12) 1 ( ) ( ) t t 1 = ( 1). (13) The positivity of the coefficients is showed using identity (13) on equation (10), σ = 1 0 [( ) ( )] s 1 + s 1 (1 s) + ds. Furtherore, using the identity (12) gives σ = (1 s)s [( ) + s 1 1 ( s 1 1 Now defining the function F (s) = ( ) +s 1 1 we have, ( ) ( ) + s 1 s 1 = F (s) F ( s) = 1 1 Perforing the differentiation we get, F (s) = F (s) (ψ( + s) ψ(1 + s)), s s )] ds. (14) Ψ (s)ds. where ψ(t) is the digaa function. Using that ψ(t) is onotonously increasing ( for t > 0 and F (s) > 0 for s > 1 and > 1 we see that +s 1 ) ( 1 s 1 ) 1 is positive, which iplies that σ > 0 as > 1. By inspection it is verified that σ 1 = 0, proving the first part of the proposition. Using equation (14) we get for the difference of two consecutive coefficients σ σ +1 = 1 0 (1 s)s (F (s) F ( s)) ds, with F (s) = 1 ( ) + s 1 ( ) + s 1 = s 1 ( ) + s ( + 1) 1 where the last equality follows fro the recursion ( ) ( t = t 1 ) ( 1 + t 1 ) and identity (12). Applying the identity once ore yields, F (s) = s 2 1 ( + 1)( 1) ˆF (s), 5

8 with such that σ σ +1 = 1 0 ( ) + s 1 ˆF (s) =, 2 (1 s)s(s 2 1) ( ) ˆF (s) ˆF ( s) ds. (15) ( + 1)( 1) Again we have ˆF (s) ˆF ( s) = s s ˆF (s)ds. Perforing the differentiation yields ˆF (s) = ˆF (s) (ψ( + s) ψ(s + 2)). Fro this we see that ˆF (s) ˆF ( s) > 0 for > 2. Together with equation (15), σ > σ +1, for > 2. By inspection it is verified that σ 3 = σ 2, proving the second part of the proposition. Secondly we need soe properties of Störer s and Cowell s ethods related to the order constants (3). Proposition 2. The error of a Störer ethod of order p 2 has an expansion of the for (3) where C p+2 = σ p C p+3 = p 2 2 σ p + σ p+1. Likewise, for the Cowell ethod of order p 4, C p+2 = σ p C p+3 = p 2 σ p + σ p+1. Proof. The for of the error and the expression for C p+2 is shown in [8]. Let us repeat the arguent here: Applying the order k+1 Störer ethod (8) to y(x) = x k+3, the error of the ethod ust necessarily be, since the order k + 2 ethod is exact for polynoials of degree k + 3, y n+1 2y n + y n 1 h 2 σ j j y = h 2 σ k+1 k+1 y = h k+3 (k + 3)!σ k+1. For the last equality we have used that x = h!. Now coparing with (3) yields the result. In order to find C p+3 we will need the identity x +1 = h ( + 1)! (x ) 2 h, (16) which is deonstrated by e.g. an induction arguent on Leibnitz rule for finite differences. Now the procedure is the sae as above. Applying the 6

9 ethod (8) to y(x) = x k+4, the error is, since the order k + 3 ethod is exact for polynoials of degree k + 4, y n+1 2y n + y n 1 h 2 σ j j y =h 2 σ k+1 k+1 y + h 2 σ k+2 k+2 y ( =h k+3 (k + 4)!σ k+1 x + k 1 ) h 2 + h k+4 (k + 4)!σ k+2. Collecting the coefficient of h k+3 and coparing with (3) gives C p+3 = σ k+2 + σ k+1 k 1 2 = p 2 2 σ p+1 + σ p. Repeating these steps in a straightforward anner yields the stated result for Cowell s ethods. 3 Absolute stability as h 0 of Störer s and Cowell s ethods In this section we shall discuss the stability of the Störer and Cowell ethods in detail. Regarding zero stability, as defined in Definition 1, it is easily seen that all Störer-Cowell ethods satisfy this criterion with a double root at 1 and all other roots at the origin. Note that the double root at 1 is necessary for consistency. We shall see in the following that under sall perturbations of q around 0 the double root of ρ(z) will possibly split up into two roots, and the absolute stability of the ethods now depend on whether these roots reain within the unit disk or not. Before discussing the absolute stability of the ethods, we shall need soe ore background on the notion of absolute stability defined in Definition 2. Considering the test equation y = λy, (17) with solutions of the for y(x) = C 1 e iλx + C 2 e iλx, we see that solutions are bounded for real λ. Applying the ethod (2) to this equation, using an ansatz of the for y(x 0 + nh) y n = ζ n leads to the characteristic equation ϕ(ζ) = ρ(ζ) q 2 σ(ζ) = 0, q = (λh) 2. (18) Clearly, a root of agnitude larger than one will lead to a possibly unbounded nuerical solution. This lies behind the definition of absolute stability in Definition 2. We will in the following analysis refer both to the test equation and characteristic equation to arrive at the result. In order to investigate the absolute stability of our ethods near zero we clearly have to focus our attention on the double root of the characteristic equation at q = 0 and deterine how it oves with growing q. Therefore we write ϕ(r(q)) = 0, r(0) = 1, and investigate the absolute value of r(q) for sall values of q. Now write r(q) = e iq + g(q), 7

10 where we note that li q 0 g(q) = 0. equation (18) and expanding gives Inserting into the characteristic ρ(e iq )+g(q)ρ (q iq )+O(g 2 )+q 2 σ(e iq )+g(q)q 2 σ (e iq )+O(q 2 g 2 ) = 0. (19) In the following we have to work with this equation for odd and even orders separately. 3.1 Odd orders Assuing the ethod is of order p = with 1, the there holds, using (3) with solutions of (17) ρ(e iq ) + q 2 σ(e iq ) = C p+2 h p+2 (iλ) p+2 + C p+3 h p+3 (iλ) p+3 + O(q p+4 ) Inserting into (19) gives = i( 1) +1 C p+2 q p+2 + ( 1) C p+3 q p+3 + O(q p+4 ). [ ρ (e iq ) + q 2 σ (e iq ) ] g(q) + O(g 2 )(1 + q 2 ) = ( 1) +1 C p+3 q p+3 + i( 1) C p+2 q p+2 + O(q p+4 ). (20) Coparing orders of q gives that g(q) = O(q p+2 ). Using the consistency of the ethod we get that, Likewise, ρ (e iq ) = ρ (1)qi 1 2 (ρ (1) + ρ (1))q 2 + O(q 3 ). (21) σ (e iq ) = σ (1)qi + Therefore we get [σ (1) 12 (σ (1) + σ (1))q 2 ] + O(q 3 ). (22) ρ (e iq ) + q 2 σ (e iq ) = a 0 q 2 + a 1 qi + O(q 3 ), (23) where a 0 = 1 2 (ρ (1) + ρ (1)) + σ (1), a 1 = ρ (1). (24) Now isolate the real and iaginary parts of g(q), g(q) = g 0 (q) + ig 1 (q), substitute into (20), and isolate real and iaginary parts of the equation(noting that C p+2 and C p+3 are real by Proposition 1), a 0 g 0 (q)q 2 a 1 g 1 (q)q = ( 1) +1 C p+3 q p+3 + O(q p+4 ), (25) a 1 g 0 (q)q + a 0 q 1 (q)q 2 = ( 1) C p+2 q p+2 + O(q p+3 ). Again coparing orders of q shows that g 0 (q) = O(q p+1 ) and g 1 (q) = O(q p+2 ). Eliinating higher order ters and solving for g 0 (q) gives, g 0 (q) = ( 1) C p+2 ρ (1) qp+1 + O(q p+2 ). (26) Now we are in position to investigate the size of r(q) under sall perturbations, r(q) = 1 + g(q) 2 + 2R[e iq g(q)] = 1 + 2g 0 (q) + O(q p+2 ). 8

11 The final step is valid since, R[e iq g(q)] = R[(1 iq +...)g(q)] = g 0 (q) g 1 (q)q + O(gq 2 ), and g 1 (q) = O(q p+2 ). The absolute stability of the ethod for sall q can thus be characterized in ters of the sign of the function A o (p, q), A o (, ρ) = 2 ( 1) C 2+3 ρ. (27) (1) This will be ade explicit in Theore 1. This theore also includes the case of even orders, which will be investigated in the following. 3.2 Even orders We proceed analogously for even orders. Assuing p = 2, 2, then the there holds, ρ(e iq ) + q 2 σ(e iq ) = C p+2 h p+2 (iλ) p+2 + C p+3 h p+3 (iλ) p+3 + O(q p+4 ) = ( 1) +1 C p+2 q p+2 + i( 1) +1 C p+3 q p+3 + O(q p+4 ). Following the steps (21), (22), (23), with g(q) = g 0 (q) + ig 1 (q), leads to the syste a 0 g 0 (q)q 2 a 1 g 1 (q)q = ( 1) C p+2 q p+2 + O(q p+3 ), (28) a 1 g 0 (q)q + a 0 g 1 (q)q 2 = ( 1) C p+3 q p+3 + O(q p+4 ). Coparing orders of q shows now that g 0 (q) = O(q p+2 ) and g 1 (q) = O(q p+1 ). Eliinating higher order ters and solving for g 0 (q) and g 1 (q) gives, g 1 (q) =( 1) +1 C p+2 ρ (1) qp+1 + O(q p+2 ), (29) [ g 0 (q) = ( 1) ρ (1) 2 ρ (1)C p+3 1 ] 2 C p+2 (ρ (1) + ρ (1) 2σ (1)) q p+2 + O(q p+3 ). Order conditions for order 2 [8] requires that ρ (1) = 2σ(1), and ρ (1) = 6σ (1) 6σ(1). (30) Thus, inserting (30) into (29), we get g 0 (q) = ( 1) ρ (1) 2 [ρ (1)C p+3 13 ρ (1)C p+2 ] q p+2 + O(q p+3 ). Again investigating the size of the roots leads to where r(q) = 1 + g(q) + 2R[e iq g(q)] = 1 + A e (, ρ)q p+2 + O(q p+3 ), A e (, ρ) = 2 ( 1) ρ (1) 2 [ ( ) ] 1 ρ (1)C ρ (1) + ρ (1) C 2+2. (31) 9

12 k Störer Cowell Table 2: Methods that are stable in the vicinity of zero: denotes stable ethod, 0 - unstable. 3.3 The result We now synthesize the ain result of this paper, the stability Störer and Cowell ethods. Theore The k-step Störer ethod of order p = k +1 (8) is absolutely stable for q 0, q 0, whenever p = 4l 1 or p = 4l, and absolutely unstable whenever p = 4l + 1 or p = 4l + 2, l = 1, 2, The k-step Cowell ethod of order p = k+1 (9) is absolutely unstable for q 0, q 0, whenever p = 4l 1 or p = 4l, and absolutely stable whenever p = 4l + 1 or p = 4l + 2, l = 1, 2,.... l = 1, 2,.... Proof. In this proof we will need the following regarding the characteristic polynoial ρ(ζ), defined in equation (7) ρ (1) = 2, and 1 3 ρ (1) + ρ (1) = 2(k 1). (32) Both equalities are easily verified by straightforward calculations. 1. For Störer s ethod we have fro Propositions 1 and 2 that C p+2 = σ p+1 > 0. Therefore, investigating the sign of A o (, ρ) defined in equation (27), using equation (32), we see that the ethod will be absolutely stable for q 0, q > 0, whenever is odd; = 2l 1, l = 1, 2,.... This corresponds to order p = = 4l 1. Likewise the ethod is unstable for even ; = 2l, l = 1, This corresponds to order p = = 4l + 1. For even orders we investigate the function A e (, ρ) defined in equation (31). Using proposition 2 and equation (32), we have for Störer s ethods, A e (, ρ) = ( 1) (C p+3 (p 2)C p+2 ) ( = ( 1) σ p+2 p 2 ) 2 σ p+1. (33) Now Proposition 1 guarantees that the factor σ p+2 (p 2) 2 σ p+1 is negative as long as p > 3. Using this, we see that the ethod is absolutely stable when q 0, q > 0, if is even, = 2l, l = 1, 2,..., corresponding to order p = 2 = 4l. Likewise will the ethod be absolutely unstable, q 0, q > 0, if p = 4l For Cowell s ethod we repeat the exact sae arguent as for Störer s, but with reversed signs. This gives that, provided p > 3, the ethod is absolutely unstable in the vicinity of zero whenever Störer s ethod is absolutely stable and vice versa. 10

13 Thus we have established the stability of the Störer s ethods and Cowell s ethods for k > 3. For saller k stability is checked case by case. This will be done ore in detail in the following section where we shall establish intervals of stability for soe of the lower order ethods. We su up the stability of the ethods in the vicinity of zero in Table 2. 4 Regions of absolute stability In order to visualize the actual regions of absolute stability we use what is known as root-locus curve in the classical theory of ultistep ethods[6]. The root-locus curve in the case of the ethod (6) is siply the iage of the unit circle under the transforation z σ(z)/ϕ(z). The iportance of this curve lies in the fact that the boundary of the region of absolute stability will necessarily be a subset of this curve (a) (b) (c) (d) Figure 1: The Root-Locus curve and stability regions for Störer s ethods: (a) k = 2, (b) k = 3, (c) k = 4, (d) k = 5. Starting with Störer s ethod, we know that k = 0 and k = 1 both correspond to the order 2 Störer-Verlet ethod. It is easily verified that this ethod is absolutely stable for q < 2. For k = 2, 3, 4, 5 we draw the root-locus curves, and deterine the stability regions case by case. Figure 11

14 k Störer Cowell 0 [ 2, 2] [0, ] 1 [ 2, 2] [0, 4] 2 [0, 3] [0, 6] 3 [0, 2] [0, 6] 4 [ , ] [0, 11 ] 5 unstable [0, ] 6 [0, ] [0.9314, 52 ] 7 [0, ] [ , 71 ] 8 unstable [0, ] 9 unstable [0, ] [0, ] [ , [0, ] unstable 12 unstable [0, ] 13 unstable [0, ] ] Table 3: Stability intervals for Störer s and Cowell s ethods. 1 shows the result of these calculations. Note that in the case k = 4 and k = 5, there is no apparent region of stability. However, in the case of k = 4 we can zoo in and verify that there is in fact a sall region of stability around q 1.2 soething that ight coe as a slight surprise, see Figure 3. For k = 5 the sae kind of investigation shows that there is indeed no regions of stability. For Cowell s ethods it can be verified in a siilar case by case investigations that the ethods k = 0, 1, 2, 3 are stable near zero. In Figure 2 we plot root-locus curves for the ethods k = 4, 5, 6, 7. In the case of Cowell k = 7, there appears to be no region of stability. Again, as in the case of Störer k = 4, by zooing in it is verified that there is in fact a sall region of stability away fro zero. For Cowell s ethod with k = 6 one can be isled by Figure 2c) to believe that the ethod is stable near zero. However, by zooing, as shown in Figure 3b) we see that it has a region of stability with lower real liit close to one. Thus, by carefully exaining case by case we can obtain real intervals of stability for higher order ethods. This is done in a nuerically satisfactory way by finding all points where the root locus curve crosses the real axis, and then test the size of all roots in between to deterine if the corresponding intervals are stable or unstable. The result for k up to 13 is listed in Table 3. References [1] R. Barrio, M. Rodríguez, A. Abad, and F. Blesa, Breaking the liits: The Taylor series ethod, Applied atheatics and coputation, 217 (2011), pp [2] G. Dahlquist, On accuracy and unconditional stability of linear ultistep ethods for second order differential equations, BIT; Nordisk Tidskrift for Inforationsbehandling (BIT), 18 (1978), pp

15 (a) 0.6 (b) (c) (d) Figure 2: The Root-Locus curve and stability regions for Cowell s ethods: (a) k = 4, (b) k = 5, (c) k = 6, (d) k = (a) (b) Figure 3: Zoo-in on stability regions for (a) Störer s ethod with k = 4, (b) Cowell s ethod with k = 7. 13

16 [3] W. Gautschi, Nuerical integration of ordinary differential equations based on trigonoetric polynoials, Nuerische Matheatik, 3 (1961), pp [4] K. Grazier, W. Newan, J. Hyan, P. Sharp, and D. Goldstein, Achieving Brouwer s law with high-order Störer ultistep ethods, ANZIAM J, 46 (2004), p. 05. [5] E. Hairer, C. Lubich, and G. Wanner, Geoetric nuerical integration illustrated by the Störer-Verlet ethod, Acta Nuerica, 12 (2003), pp [6] E. Hairer, S. Nørsett, and G. Wanner, Solving Ordinary Differential Equations: Nonstiff Probles, vol. 1, Springer Verlag, [7] W. Hayes, Surfing on the edge: chaos versus near-integrability in the syste of jovian planets, Monthly Notices of the Royal Astronoical Society, 386 (2008), pp [8] P. Henrici, Discrete variable ethods in ordinary differential equations, vol. 1, New York: Wiley, 1962, [9] J. Labert and J. Labert, Coputational Methods in Ordinary Differential Equations, Wiley New York, [10] J. Labert and I. Watson, Syetric ultistip ethods for periodic initial value probles, IMA Journal of Applied Matheatics, 18 (1976), p [11] W. I. Newan, F. Varadi, A. Y. Lee, W. M. Kaula, K. R. Grazier, and J. M. Hyan, Nuerical integration, Lyapunov exponents and the outer Solar Syste, Bulletin of the Aerican Astronoical Society, 32 (2000), p [12] F. Olver, D. Lozier, R. Boisvert, and C. Clark, NIST handbook of atheatical functions, Cabridge University Press New York, NY, USA, [13] G. Quinlan and S. Treaine, Syetric ultistep ethods for the nuerical integration of planetary orbits, The Astronoical Journal, 100 (1990), pp [14] E. Thorbergsen, Undersøkelse av noen etoder for baneprobleer, Master s thesis, Norges Tekniske Høyskole(NTH), Trondhei, Norway, [15] F. Varadi and B. Runnegar, Successive refineents in long-ter integrations of planetary orbits, The Astrophysical Journal, (2003). 14