THE ASTRONOMICAL JOURNAL, 120:3333È3339, 2000 December ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. LONG-TERM INTEGRATION ERROR OF KUSTAANHEIMO-STIEFEL REGULARIZED ORBITAL MOTION HIDEYOSHI ARAKIDA Department of Astronomical Science, Graduate University for Advanced Studies, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan; h.arakida=nao.ac.jp AND TOSHIO FUKUSHIMA National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan; Toshio.Fukushima=nao.ac.jp Received 2000 April 18; accepted 2000 July 31 ABSTRACT We conðrm that the positional error of a perturbed two-body problem expressed in the Kustaanheimo-Stiefel (K-S) variable is proportional to the Ðctitious time s, which is the independent variable in the K-S transformation. This property does not depend on the type of perturbation, on the integrator used, or on the initial conditions, including the nominal eccentricity. The error growth of the physical time evolution and the Kepler energy is proportional to s if the perturbed harmonic oscillator part of the equation of motion is integrated by a time-symmetric integration formula, such as the leapfrog or the symmetric multistep method, and is proportional to s2 when using traditional integrators, such as the Runge-Kutta, Adams, Sto rmer, and extrapolation methods. Also, we discovered that the K-S regularization avoids the step size resonance/instability of the symmetric multistep method that appears in the unregularized cases. Therefore, the K-S regularized equation of motion is useful to investigate the long-term behavior of perturbed two-body problems, namely, those used for studying the dynamics of comets, minor planets, the Moon, and other natural and artiðcial satellites. Key words: celestial mechanics, stellar dynamics È methods: numerical 1. INTRODUCTION For the long-term study of orbital motions, it is desirable to adopt a formulation producing small positional errors, especially those errors whose growth is proportional to time t. One e ective direction is to use a numerical integrator leading to a small growth in the positional error. Unfortunately, the traditional integrators, such as the Runge- Kutta, Adams, Sto rmer-cowell, and the extrapolation methods, are not suitable, in the sense that the positional error grows in proportion to t2. This error growth has been the barrier to long-term integrations of dynamical systems (Milani & Nobili 1988, Table 1). Recently, two new integrators were found to overcome this barrier: the symplectic integrator (see review by Yoshida 1995) and the symmetric multistep method (Lambert & Watson 1976; Quinlan & Tremaine 1990; Fukushima 1999; Evans & Tremaine 1999). Both produce positional errors that are proportional to time t. On the other hand, the K-S regularization (see Stiefel & Scheifele 1971) has been used to treat the two-body close approach properly. It especially reduces the accumulation of numerical error in highly eccentric orbits of comets and minor planets (Yabushita & Tsuzii 1989; Shefer 1990; Zheng 1994) and of artiðcial-satellite motion (Sharaf, Awad, & Banaja 1987). Also, it has been an essential tool in studying general N-body problems and stellar dynamics (Aarseth & Zare 1974; Mikkola & Asrseth 1990, 1993). So far, the regularization has been regarded as less e ective in the case of circular and low-eccentricity orbits. The reason is that it cannot decrease the numerical error signiðcantly but requires a signiðcant increase in computational time and complexity of the coding. However, is it the only advantage of the K-S regularization to avoid the close approach of celestial bodies? Is there not another beneðt in the fact that the K-S regularization transforms a nonlinear perturbed two-body problem into a 3333 linear perturbed harmonic oscillator? In this paper, we will report that this fact greatly reduces the long-term integration error of the orbital motion. In 2, we investigate the numerical integration error of the K-S regularized equation of motion. In 3, we present the fact that the K-S regularization avoids the step size resonance/instability of the symmetric multistep method, which appears in the unregularized cases as reported by Quinlan (1999) and Fukushima (1999). 2. KOLMOGOROV-SMIRNOV REGULARIZED ORBITAL MOTIONS The main part of K-S regularized equation of motion is written in the form of a four-dimensional perturbed harmonic oscillator (Stiefel & Scheifele 1971; Brumberg 1995). Its independent variable is not the physical time but the Ðctitious time s, which is a variation of an eccentric anomaly. A simple error analysis shows that the positional error of the K-S equation of motion is expected to grow only in proportion to s. This is because the frequency u is hardly independent of the energy integrated, and therefore, the phase error, being an integral of the frequency error, is proportional to s. On the other hand, the discussion in Appendix A concludes that the error of physical time generally contains terms of both the odd and even powers of the step size, h. The terms of odd powers grow in proportion to s2, while the terms of even powers grow in proportion to s. Also, it is easily shown that only the terms of even powers of h remain in the cases of the time-symmetric integrators, such as the leapfrog and the symmetric multistep methods. Therefore, it is expected that the error of time evolution grows in proportion to s2 for the traditional integrators and in proportion to s for the time-symmetric integrators. Now let us examine these conjectures by conducting numerical experiments. We adopted four types of simple orbital motions as test problems: (1) the pure Keplerian
3334 ARAKIDA & FUKUSHIMA Vol. 120 problem (denoted by K hereafter), (2) the perturbed Keplerian orbit under the e ect of air dragging force (A), (3) the circular restricted, three-dimensional three-body problem (R3), and (4) the general three-dimensional threebody problem (G3). As for the air dragging force, we selected the following form: F \ [c D A A 2m ov2 (1) v (Vallado 1997, eq. [7-24]), where c is the coefficient of drag, o is the atmospheric density, D and A and m are the cross-sectional area and the mass of the artiðcial satellite, respectively. In the circular restricted and the general threebody problems, we named the bodies Sun, Jupiter, and Asteroid. The adopted initial conditions for Jupiter and Asteroid are listed in Table 1. We regarded the mass of Jupiter as m \ 10~3 M and that of Asteroid as m \ 0 (restricted) and J m \ 10~7 _ M (general), respectively, A where M is the mass A of Sun and _ the gravitational constant is set as G _ \ 1. Figure 1 shows the perturbation type dependence of the positional error growth while the integrator was Ðxed as the fourth-order Adams-Bashforth method. The errors are plotted as functions of the independent variable s. See Appendices B and C for the details of the integrators we used and the deðnition of errors, respectively. Figure 2 illustrates the integrator dependence of the positional error growth, where the perturbation was Ðxed as the restricted three-body problem. In all the cases, the positional error growth was proportional to the Ðctitious time s. Table 2 lists the comparison of magnitudes of positional error among the same order of integrators, while the number of the function calls was set the same for all the integrators. Next let us consider the eccentricity dependence in the K-S transformation. See Figure 3 for the eccentricity dependence of the positional error after 120 nominal revolutions, where the integrator and problem are Ðxed as the eighthorder symmetric multistep method (SM8) and pure Kepler motion, respectively. In the K-S regularized case, the error hardly depends on the eccentricity. In the unregularized case, however, the error drastically increases according to the eccentricity. This is because the eccentricity dependence FIG. 1.ÈPerturbation type dependence of the positional error growth of the K-S regularized perturbed two-body problem. We adopted the fourth-order Adams-Bashforth method (AB4) as the integrator. The step size was Ðxed as h \ 0.06, where the nominal orbital period is 2n. The errors were plotted after multiplying by some constant factors to make the di erence clear: 100.5 for the perturbation due to the air dragging force (A), 101.0 for the restricted three-body problem (R3), and 101.5 for the general three-body problem (G3). appears as the initial phase of the corresponding harmonic oscillator in the K-S regularization. In conclusion, this phenomenonèthat the positional error growth is proportional to sèis independent of the perturbations, the integration methods, and the initial conditions, including the nominal eccentricity. Now examine the other errors: physical time, Kepler energy, and conserved quantities. Figure 4 shows the integrator dependence of the error of time evolution of the pure TABLE 1 ADOPTED INITIAL CONDITIONS FOR THREE-BODY PROBLEMS a I ) u l Object (AU) e (deg) (deg) (deg) (deg) Asteroid... 1 0.05 20 20 20 20 Jupiter... 5.2 0.0 0 0 0 0 NOTE.ÈElements shown are referred to the Sun. FIG. 2.ÈSame as Fig. 1, but the integrator for the main K-S variables has been changed, while the integrator for the other variables, t and h, is k still Ðxed as AB4. The perturbation type is Ðxed as the circular restricted three-body problem. The line AB6 is almost the same as the line of RKF8, so the di erence is hardly seen in the Ðgure. TABLE 2 MAGNITUDE OF THE POSITIONAL ERROR OF THE PERTURBED TWO-BODY PROBLEM Order of Integration Fourth... Sixth... Eighth... Magnitude SM4] AB4 \ S4 ] AB4 \ ET4 ] AB4 \ AB4 \ RK4 \ EX4 SM6] AB6 \ S6 ] AB6 \ AB6 \ EX6 SM8] AB6 \ S8 ] AB8 \ AB8 \ RKF8 NOTE.ÈThe magnitudes of the positional error are compared with respect to each order of integration when the number of the function calls per unit time are the same.
No. 6, 2000 K-S REGULARIZED ORBITAL MOTION 3335 FIG. 3.ÈEccentricity dependence of the positional error. The eccentricity dependence is compared with the K-S regularized and unregularized approaches. The problem is Ðxed as a pure Kepler problem. The errors after 120 revolutions are plotted as a function of the eccentricity. The integrator is Ðxed as the eighth-order symmetric multistep method of Quinlan & Tramaine (1990). Kepler problem. We obtained similar Ðgures even when perturbations existed. Stiefel (1970) found that the error of physical time propagates in proportion to s in the case of the fourth-order Runge-Kutta method (RK4). However, as we considered at the beginning of this section, the error of time evolution grows in proportion to s2 for the traditional integrators, including RK4, and grows in proportion to s for the symmetric multistep methods. This is natural since the origin of the quadratic error growth is not in the truncation error of the Ðrst-order di erential equations but in the nonzero averaged value of the right-hand side of the variational equation. As a result, if the traditional integrators were used in the restricted three-body problem, the posi- FIG. 4.ÈSame as Fig. 2, but for the error growth of physical time t, integrating the part of perturbed harmonic oscillator by using both the traditional integrators and time-symmetric integrators. The problem is Ðxed as a pure Kepler problem. FIG. 5.ÈSame as Fig. 2, but for the error of the Jacobi integral tional error of Asteroid would grow only in proportion to s, while that of Jupiter, the perturber, would grow in proportion to s2. This is because the position of Jupiter in evaluating perturbations is calculated by way of the integrated physical time. We also conðrmed that the error growth of the Kepler energy h is the same as the physical time t. The K error growth of the conserved quantities is proportional to s for the traditional integrators and remains constant for the symmetric integrators. See Figure 5 for the case of the Jacobi integral in the circular restricted three-body problem. The same phenomenon was seen in the total energy and angular momentum of Kepler motion. Table 3 is the same as Table 2, but it compares the magnitude of errors in the Jacobi integral. Finally, we discuss the applicability of this approach to general N-body problems. Since the Ðctitious time s is proper to each body, there exist N di erent Ðctitious times in general N-body problems. Thus, one has to choose a speciðc s or a suitable function of them as the independent variable. As such, we adopted the Ðctitious time of Asteroid in the case of the general three-body problem. In this case, the equation of motion of the K-S variables of Jupiter di ers from that of perturbed harmonic oscillator since the additional factor r /r appears and varies with time. As a A J result, the equation of motion of Jupiter in the K-S variable is no longer expressed in the form of a perturbed harmonic oscillator. This is the reason that the positional error growth of the perturber, Jupiter, is proportional to s2 for the Runge-Kutta and Adams-Bashforth methods. Note that this situation in unchanged when the independent variable is chosen in another way. In other words, the present approach does not work well in general N-body problems. 3. REDUCTION OF STEP SIZE RESONANCE/INSTABILITY IN THE SYMMETRIC MULTISTEP METHOD As Quinlan (1999) and Fukushima (1999) have shown, some higher order symmetric multistep methods for the TABLE 3 MAGNITUDE OF ERROR OF CONSERVED QUANTITIES OF THE PERTURBED TWO-BODY PROBLEM Order of Integration Fourth... Sixth... Eighth... Magnitude SM4] AB4 \ ET4 ] AB4 > S4 ] AB4 \ AB4 \ RK4 \ EX4 SM6] AB6 > S6 ] AB6 \ AB6 \ EX6 SM8] AB8 > S8 ] AB8 \ AB8 \ RKF8
3336 ARAKIDA & FUKUSHIMA Vol. 120 special second-order ordinary di erential equations (ODEs) face the step size resonance/instability when they are applied to nonlinear ODEs, such as the Kepler problem. Quinlan (1999) explained the mechanism of such resonance/ instability by a linear stability analysis of a planar orbital motion under an axisymmetric potential /(r). He concluded that the stability condition in terms of the dispersion relation is u (r) \ 1[4u2(r) [ i2(r)] \ 0, (2) 2 2 where u(r) is the circular frequency, deðned as u2(r) \ /@(r)/r, (3) and i(r) is an epicycle frequency deðned as i2(r) 4 r du2 dr ] 4u2\/@@(r) ] 3 /@(r). (4) r The solution of the di erential equation (eq. [2]) with respect to / is easily obtained as /(r) \ u2 r2]/, (5) 0 0 where u and / are the integration constants. Thus, we 0 0 have shown that the stable form of /(r) is essentially limited to that of a harmonic oscillator. Therefore, harmonic oscillation is the only case in which the step size instability does not appear. Let us present the proof by numerical experiments. Figure 6 illustrates the step size dependence of the energy error of the one-dimensional perturbed harmonic oscillator. The Ðgure shows the error after 1000 revolutions, where the perturbing force is of the form F \ vx. We used the nthorder (n \ 2, 4, 6, 8, 10, and 12) symmetric multistep method as the integrator for the special second-order ODEs (Lambert & Watson 1976; Quinlan & Tremaine 1990) and the fourth-order symmetric multistep method for the general Ðrst-order ODEs (Evans & Tremaine 1999). Note that they show eminent step size resonance/instabilities in integrating a Kepler problem (Quinlan 1999; Fukushima FIG. 6.ÈStep size dependence of the energy error of a perturbed harmonic oscillator. Illustrated is the relative energy error after 1000 nominal revolutions as a function of steps per period. The integrators adopted are the symmetric multistep methods. We denote the leapfrog method by LF and method IV of Lambert & Watson (1976) by SM4; SM6 is method VI of Lambert & Watson (1976), where the free parameter a was set as a \ 0; the 8th-, 10th-, and 12th-order formulas of Quinlan & Tremaine (1990) are denoted by SM8, SM10, and SM12, respectively, and the fourth-order formula of Evans & Tremaine (1999), where the free parameter was set as u \ 0, by ET4. 2 FIG. 7.ÈSame as Fig. 6, but the errors of Jacobi integral after 104 nominal revolutions are plotted as functions of the steps per period. The integrator is Ðxed as the eighth-order symmetric multistep method of Quinlan & Tremaine (1990). The initial eccentricity is Ðxed as e \ 0.2. 1999). Obviously, no resonance/instability appears in Figure 6. Next, Figure 7 shows the step size dependence of the error of the Jacobi integral of the circular restricted three-body problem after 10,000 nominal revolutions. Here we Ðxed the integrator as the eighth-order symmetric multistep method of Quinlan & Tremaine (1990) both for the K-S regularized and unregularized equations of motion. In the regularized case, the error decreases monotonically with respect to the step size, while many spikes appear in the unregularized case. Thus, we have numerically conðrmed that the step size resonance/instability in the symmetric multistep methods is avoided by transforming the Kepler problem into the K-S form. 4. CONCLUSION We have investigated the long-term behavior of the integration errors of K-S regularized orbital motions. Numerical experiments show that the K-S regularized equation of motion is superior to the ordinary equation of motion in rectangular coordinates in the sense that their numerical integrations lead to signiðcantly smaller errors in position. This property hardly depends on the type of the perturbation considered, on the integrators used, or on the initial conditions adopted, especially on the value of nominal eccentricity. It is known that some symmetric multistep methods for the special second-order ODEs have a curious instability with respect to the magnitude of the step size in integrating orbital motions in ordinary rectangular coordinates (Quinlan 1999; Fukushima 1999). However, from the results in 3, we conðrm that this feature disappears in the perturbed harmonic oscillator and, therefore, in K-S regularized orbital motions. Hence, the symmetric multistep method seems the most appropriate to integrate the K-S regularized orbital motion because it achieved the highest cost performance among the integrators tested. Of course, the most practical concern in applying the K-S regularization would be the increase in computational time. Figure 8 shows a comparison of the CPU time of numerical integrations with the K-S regularized and unregularized equations of motion, plotted as a function of the number of perturbing bodies. In the case of no perturbing bodies, the CPU time of the regularized case is about 40% larger than that of the unregularized one. However, the CPU time for
No. 6, 2000 K-S REGULARIZED ORBITAL MOTION 3337 FIG. 8.ÈComparison of CPU time. Illustrated is the ratio of CPU times of numerical integration of one step between the K-S regularized and unregularized equations of motion. The CPU times are plotted as a function of the number of perturbing bodies. the regularized case becomes about 10% larger than that of the unregularized case for two perturbing bodies, and about only a few percent larger for more than six perturbing bodies, so that the di erence in CPU time between the regularized case and the unregularized one almost reduces. Therefore, although the K-S regularization requires the integration of 10 variables instead of six in rectangular coordinates, the actual increase in CPU time is not signiðcant if the force computation is sufficiently complicated, say, more than that of six perturbers in the case of asteroid integration, or more than a three-degree or higher gravitational potential of Earth in the case of an artiðcial satellite. In conclusion, the K-S regularization is useful not only because it properly deals with close approaches among the celestial bodies (see, e.g., Fig. 3), but because the positional error growth is only proportional to the Ðctitious time s. The regularization also stabilizes the numerical integration by the symmetric multistep method for the special secondorder ODEs, which turns out to be the most efficient integrator. However, this good property fails in the general N-body (N º 3) problem because the Ðctitious time is proper to each body. Therefore, the K-S regularization is e ective to study the long-term behavior of perturbed twobody problems, especially the dynamics of comets, minor planets, the Moon, and natural and artiðcial satellites, including the L aser Geodynamic Satellite and GPS/Navstar. The authors would like to thank the referee for fruitful comments and suggestions. APPENDIX A ERROR GROWTH OF THE PHYSICAL TIME EVOLUTION In K-S regularization, the physical time evolution is obtained by numerical integration. Let us consider its error growth. To simplify the situation, we restrict ourselves to its two-dimensional subset, the Levi-Civita transformation (Stiefel & Scheifele 1971, p. 20). Without losing generality, we assume the transformed equation of motion is written as d2u k ds2 ] u \ 0 (k \ 1, 2) (A1) k under the initial conditions u 1 (0) \ 1, u 2 (0) \ 0, du (0) 1 \ 0, ds du (0) 2 \ 1, ds where u are new variables connected to the position vector (x, y) as k x \ u2[u2, and y \ 2u u. The analytical solution of equation (A1) 1 is 2 1 2 u \ cos s, u \ sin s. (A2) 1 2 Consider the truncation error of the numerical integration. According to Henrici (1962), the di erential equation of error, *u, is given as k A d*u2 k ds2 ] *u k \ ; = Cj hp d uk (k \ 1, 2). (A3) dsbp j/p This can be rewritten as d2 (*u t 1 ) t; ] ( *u t 1 ) t; \ ( C C t even odd )( cos s) t; t: t ds2 :*u 2 : *u 2 : [C C sin s odd even ;, (A4) where = C \ ; C2i h2i([1)i, even i/kp@2l = C \ ; C2i~1 h2i~1([1)i, (A5) odd i/kp@2l and C are the error constants proper to the integrator. In j general, if the initial conditions are given by (*u d *u /ds t 1 1 ) t :*u d *u /ds 2 2 ; \ ( a b t 1 1 ) t; (s \ 0), (A6) : a b 2 2 then the solutions of equation (A4) become (*u t 1 ) t; \ 1 ( C C t even odd )( s sin s ) t; t: t :*u 2 2 :[C C sin s [ s cos s odd even ; ] ( a b t 1 1 )( cos s) t; t: t : a b sin s 2 2 ;. (A7) The di erential equation describing the time evolution is dt ds \ r \ u 1 2]u 2, (A8) whence the variational equation becomes d *t ds \ 2(u 1 *u 1 ] u 2 *u 2 ). (A9) This equation is explicitly integrated as As sin 2sB As2 cos 2s *t \ C [ [ Codd B ] even 2 4 2 4 C ]2 (a ] b ) s 1 2 2 [ (b 2 ] a cos 2s ) 1 2 sin 2sD ] (a [ b ). 1 2 4 (A10) In general, there appear error terms of both odd and even powers of h. Thus, *t grows in proportion to s2. However,
3338 ARAKIDA & FUKUSHIMA Vol. 120 in the case of the time-symmetric integrators, *u contains only the even powers of h. Therefore, the error k *t only grows linearly. Even if there are errors corresponding to the initial values, its contribution only appears as a linear growth. APPENDIX B ADOPTED INTEGRATORS The integrators used in this work are listed in Table 4 and are as follows: (1) the fourth-order Runge-Kutta method (RK4), (2) the eighth-order Runge-Kutta-Fehlberg method (RKF8), (3) the nth-order Adams-Bashforth method1 (ABn), (4) the nth-order GraggÏs extrapolation method (EXn), (5) the pair of the nth-order Sto rmer2 and the Adams- Bashforth methods (Sn ] ABn) (see Hairer et al. 1987 for details on these methods), (6) the pair of the nth-order symmetric multistep method for special second-order ODEs (Quinlan & Tremaine 1990) and the Adams-Bashforth method (SMn ] ABn), and (7) the similar pair of the symmetric multistep method for general Ðrst-order ODEs (Evans & Tremaine 1999) and the Adams-Bashforth method (ET4 ] AB4). In the last three cases, Sn ] ABn, SMn ] ABn, and ET4 ] AB4 mean that Sn, SMn, or ET4 was used for integrating the part of perturbed harmonic oscillators, namely, the equation of motion for the K-S variable u, while ABn was used for integrating the rest, i.e., the physical time t and the Keplerian energy h. When ET4 was K used for integrating the Kepler energy, we faced an instability. Therefore, we always adopted ABn for the rest even when the perturbed harmonic oscillator part was integrated by ET4. We adopted the predictor formula only for the multistep methods, namely, the Adams-Bashforth, Sto rmer, and symmetric multistep methods. For the extrapolation ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ 1 There are some typographic errors in the coefficients for the multistep methods listed in Stiefel & Scheifele (1971, p. 133). The correct values are as follows: a \ 5257/17,280, a \ 25,713/89,600, a \ 26,842,253/94,800,320, and 7 a \ 4,777,223/17,418,240 9 for the Adams- 10 Bashforth method; a \ 3/160, 11 a \ 275/24,192, a \ 8183/1,036,800, and a \ 4671/788,480 for 5 the Adams-Moulton 7 method. 9 11 2 Note that the order, p, of the multistep method is equivalent to the number of steps, k. However, in the Sto rmer-cowell linear multistep method, the order is usually deðned as p \ k ] 1. We adopted the deðnition of the order of the linear multistep method as p \ k in all cases. method, we Ðxed not only the step size but also the order by Ðxing the number of the extrapolation stages. For the sixthorder symmetric multistep method for special second-order ODEs (Lambert & Watson 1976, method IV), we set the free parameter a \ 0, and for the eighth-order one, we adopted the coefficients given in Quinlan & Tremaine (1990). For the fourth-order symmetric multistep method for general Ðrst-order ODEs (Evans & Tremaine 1999), we set the free parameter u \ 0. The starting values for the multistep methods were obtained 2 by the analytical solution if it existed, and by using RK4 with a sufficiently small step size otherwise. However, in the calculation of Figure 7, which is explained in 3, the starting values were given by using the variable-order extrapolation method with the error tolerance set to 10~14. Note that the second-order Sto rmer method (S2), the second-order symplectic integrator (though we did not use the symplectic integrator in this paper), and the second-order symmetric multistep method for special second-order ODEs (SM2) are equivalent to each other, i.e., the leapfrog method (LF). All the calculations were carried out in double-precision arithmetic (53 bit mantissa) by a PC with an Intel Pentium II 350 MHz CPU under Linux. In preparing the initial condition of K-S variables, we always set u \ 0. 4 APPENDIX C DEFINITION OF ERRORS We investigated the growth of positional errors for all the cases. We also examined the error growth of some quantities such as the Jacobi integral, the total energy, and the total angular momentum when they were conserved. In the pure Kepler problem, we evaluated the error by the deviation from the analytical solution: *X \ X [ X, numerical analytical where the analytical solution was that given in the explicit function of the Ðctitious time. In the other cases when the analytical solutions are not available easily, we conducted the numerical integrations by Ðxing step sizes and evaluated the error by a di erence between two runs of numerical integrations with di erent step sizes, *X \ X [ X. h1/h h2/h@2 TABLE 4 NUMERICAL INTEGRATORS TESTED FOR THE PERTURBED TWO-BODY PROBLEM PERTURBATION TYPE INTEGRATOR K R3 A G3 Adams-Bashforth (ABn)... Y Y Y Y Extrapolation (EXn)... Y Y Y... Fourth-order Runge-Kutta (RK4)... Y Y Y Y Eighth-order Runge-Kutta-Fehlberg (RKF8)... Y Y Y... Symmetric multistep (Evans & Tremaine) and Adams-Bashforth (ET4 ] AB4)... Y Y Y... Sto rmer and Adams-Bashforth (Sn ] ABn)... Y Y...... Symmetric multistep and Adams-Bashforth (SMn ] ABn)a... Y Y...... NOTE.ÈThe Sto rmer and the symmetric multistep method (Quinlan & Tremaine 1990) are not suitable for the case of the general relativistic one-body problem and the air dragging force, since the perturbation depends on the velocity. Though the formula of Evans & Tremaine (1999) is for general Ðrst-order ODEs, it shows an instability when integrating the Kepler energy h. Therefore, we adopt it only for integrating the part of the perturbed harmonic oscillator. a Quinlan & Tremaine 1990. K
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