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Efficient Numerical Solution of Orbital Problems with the use of Symmetric Four-step Trigonometrically-fitted Methods.

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Appl. Num. Anal. Comp. Math. 1, No. 1, 216 – 222 (2004) / DOI 10.1002/anac.200310018
Efficient Numerical Solution of Orbital Problems with the use of
Symmetric Four-step Trigonometrically-fitted Methods
G. Psihoyios∗1 and T.E. Simos∗∗∗∗∗2
1
2
Department of Mathematics, School of Applied Sciences, Anglia Polytechnic University, East Road, Cambridge CB1 1PT, United Kingdom.
Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR-221 00 Tripolis, Greece.
Received 30 June 2003, revised 30 October 2003, accepted 2 December 2003
Published online 15 March 2004
Key words Exponential fitting, multistep methods, finite difference methods, orbital problems.
Subject classification 65L05
An explicit hybrid symmetric four-step method of algebraic order six is developed in this paper. Numerical
comparative results from the application of the new method to well known periodic orbital problems, demonstrate the efficiency of the method presented here.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
Much research has been done for the approximate solution of second order differential equations of the form
y (t) = f (t, y(t)),
(1)
i.e. differential equations for which the function f is independent from the first derivative of y. Some of the most
frequently used methods for the numerical solution of problem (1) are the symmetric multistep methods.
Symmetric multistep methods were first presented by Lambert and Watson [1]. In [1] they show that the interval of periodicity of symmetric multistep methods is non vanishing, which reassures the existence of periodic
solutions in it [the interval of periodicity is determined by the application of the symmetric multistep method to
the test equation y (t) = −q 2 y(t). If q 2 h2 ∈ (0, T02 ), where h is the step length of the integration, then this
interval is called interval of periodicity]. Lambert and Watson [1] developed symmetric multistep methods, that
were orbitally stable (when the number of steps exceeds two). Orbital instability is a property which was presented for the family of Störmer-Cowell multistep methods, used for the solution of (1). The class of numerical
methods that is frequently used for long term integration of planetary orbits is that of symmetric multistep methods (see [2] and references therein). Quinlan and Tremaine [2] have constructed high order symmetric methods,
based on the work of Lambert and Watson.
We note here that the linear symmetric multistep methods, developed by Lambert and Watson [1] and by
Quinlan and Tremaine (see [2] and [3]), are much simpler than the hybrid (Runge-Kutta type) ones. For the long
time integration of initial value problems with oscillating solutions, these methods are very important due to their
simplicity and accuracy (especially for orbital problems).
In this paper, we develop a two-stage trigonometrically-fitted and exponentially-fitted symmetric multistep
method and we also discuss in some detail its stability characteristics. We present our numerical results by
∗
Corresponding author: e-mail: g.y.psihoyios@apu.ac.uk, Phone: +44 1223 363 271 ext. 2173, Fax: +44 1223 515 349.
Active Member of the European Academy of Sciences and Arts; Visiting Professor, Anglia Polytechnic University, Cambridge, UK
∗∗∗ e-mail: tsimos@mail.ariadne-t.gr, Phone: +30 2710 372 223, Fax: +30 210 942 0091. Please use the following address in all
correspondence: Dr T.E. Simos, Menelaou 26, Amphithea - Paleon Faliron, GR-175 64 Athens, Greece.
∗∗
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com
217
applying the new method to some established orbital type problems. For comparison purposes, we also use
certain other well known methods that can be found in the literature.
2 The New Trigonometrically-Fitted Method
Consider the the following general four-step formula:
y n+2 = 2 yn+1 − 2 yn + 2 yn−1 − yn−2 =
= h2 [b0 (fn+1 + fn−1 ) + b1 fn ],
(2)
yn+2 − 2 yn+1 + 2 yn − 2 yn−1 + yn−2 =
2
= h [c0 (f n+2 + fn−2 ) + c1 (fn+1 + fn−1 ) + c2 fn ],
(3)
where yn±i = y(x ± ih), i ∈ {0, 1, 2} fn±i = y (x ± ih), i ∈ {0, 1, 2} h is the step size, bi , i ∈ {0, 1} and
ci , i ∈ {0, 1, 2} are the parameters of the method, in order to be trigonometrically-fitted.
For the first stage of the method and for the trigonometrically-fitted case, we demand to integrate exactly any
linear combination of the functions:
{ 1, x, x2 , x3 , sin(±v x), cos(±v x) }.
(4)
For the second stage of the method and for the trigonometrically-fitted case, we demand to integrate exactly
any linear combination of the functions:
{ 1, x, x2 , x3 , x4 , x5 , sin(±v x), cos(±v x) }.
(5)
In order the coefficients of the method (bi , i ∈ {0, 1} and ci , i ∈ {0, 1, 2}) to satisfy the requirements (4)-(5),
the following system of equations must hold:
2 + 2 cos(2 w) − 4 cos(w) = −w2 (2 b0 cos(w) + b1 )
4 = 4 b0 + 2 b 1
4 cos(w) (cos(w) − 1) = −w2 (2 c0 cos(2 w) + 2 c1 cos(w) + c2 )
4 = 4 c0 + 4 c1 + 2 c2
28 = 96 c0 + 24 c1
(6)
The solution of the above system of equations is given by:
−1 − cos(2 w) + 2 cos(w) − w2
w2 cos(w) − w2
2 w2 cos(w) + 2 + 2 cos(2 w) − 4 cos(w)
b1 =
w2 cos(w) − w2
−6 cos(2 w) − 6 + 12 cos(w) − 7 w2 cos(w) + w2
c0 =
6 w2 cos(2 w) − 24 w2 cos(w) + 18 w2
7 w2 cos(2 w) + 17 w2 + 24 cos(2 w) + 24 − 48 cos(w)
c1 =
6 w2 cos(2 w) − 24 w2 cos(w) + 18 w2
2
−w cos(2 w) − 17 w2 cos(w) − 18 cos(2 w) − 18 + 36 cos(w)
c2 =
3 w2 cos(2 w) − 12 w2 cos(w) + 9 w2
b0 =
(7)
The behavior of coefficients of the first stage of the method is presented in F igure 1.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
218
G. Psihoyios and T.E. Simos: Symmetric Four-step Trigonometrically-fitted Methods
Behavior of the coefficient b[0]
Behavior of the coefficient b[1]
0
100000
-50000
80000
-100000
60000
-150000
40000
20000
-200000
0
-5
5
0
w
10
-10
Behavior of the coefficient c[0]
-5
0
w
5
10
Behavior of the coefficient c[1]
0
2.5e+10
-1e+09
2e+10
-2e+09
1.5e+10
-3e+09
-4e+09
1e+10
-5e+09
5e+09
-6e+09
0
-10
-5
5
0
w
-10
10
-5
0
w
5
10
Behavior of the coefficient c[2]
0
-1e+10
-2e+10
-3e+10
-4e+10
-10
-5
0
w
5
10
Fig. 1 Behavior of the coefficients b[i] = bi , i ∈ [0, 1] and c[j] = cj , j ∈ [0, 2] of the new method.
The above parameters converted into their Taylor series expansions are given below.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com
219
3 2
7
47
23
37
−
w +
w4 −
w6 +
w8 + . . .
6 40
15120
604800
39916800
3 2
1
47
23
37
w −
w4 +
w6 −
w8 + . . .
b1 = − +
3 20
7560
302400
19958400
19
3
83
1
577
+
w2 +
w4 +
w6 −
w8 + . . .
c0 =
40 6048
907200
591360
93405312000
19
13
83
1
577
−
w2 −
w4 −
w6 +
w8 + . . .
c1 =
15 1512
226800
147840
23351328000
19
7
83
1
577
+
w2 +
w4 +
w6 −
w8 + . . .
c2 =
(8)
60 1008
151200
98560
15567552000
Based on the above coefficients we can find that the local truncation error of the above scheme (2)-(3) is given
by:
b0 =
L.T.E(h) = −
1
h8 (950 yn(8) + 1701 yn(6) + 950 w2 yn(6) + 1701 w2 yn(4)
302400
(9)
3 Numerical Illustrations
3.1 A problem by Franco and Palacios
Consider the ”almost” periodic orbit problem studied in [4]:
y + y = eiψx , y(0) = 1, y (0) = i, y ∈ C ,
(10)
that has an equivalent form:
u + u = cos(ψx), u(0) = 1, u (0) = 0,
(11)
v + v = sin(ψx), v(0) = 0, v (0) = 1,
(12)
where = 0.001 and ψ = 0.01.
The analytical solution of problem (10) is given below:
y(x) = u(x) + i v(x), u, v ∈ R,
(13)
1 − − ψ2
cos(x) +
cos(ψx),
2
1−ψ
1 − ψ2
(14)
1 − ψ − ψ2
sin(x) +
sin(ψx).
1 − ψ2
1 − ψ2
(15)
u(x) =
v(x) =
The solution of the equations (13)-(15) represents motion of a perturbation of a circular orbit in the complex
plane.
For comparison purposes in our numerical illustration we use the well known eighth algebraic order method
developed by Quinlan and Tremaine [2] (which is indicated as Method [a]) and the new trigonometrically-fitted
method (2)-(3) developed in this paper is indicated as method [b].
The numerical results obtained for the two methods, with stepsizes equal to h = 21n , were compared with the
analytical solution. F igure 2 shows the errors:
Errmax = log10 max0≤x≤100000 |u, v calculated (x) − u, v theoretical (x)|
(16)
for several values of n.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
220
G. Psihoyios and T.E. Simos: Symmetric Four-step Trigonometrically-fitted Methods
0
Method [a]
Method [b]
-2
Err
-4
-6
-8
-10
-1
0
1
2
n
Fig. 2 Error Errmax for several values of n for the problem of Franco and Palacios [4].
3.2 A problem by Stiefel and Bettis
Consider the ”almost” periodic orbit problem studied in [5].
y + y = 0.001 eix , y(0) = 1, y (0) = 0.9995 i, y ∈ C ,
(17)
with an equivalent form of:
u + u = 0.001 cos(x), u(0) = 1, u (0) = 0,
(18)
v + v = 0.001 sin(x), v(0) = 0, v (0) = 0.9995.
(19)
The analytical solution of problem (17) is:
y(x) = u(x) + i v(x), u, v ∈ R,
(20)
u(x) = cos(x) + 0.0005 x sin(x),
(21)
v(x) = sin(x) − 0.0005 x cos(x).
(22)
The solution of equations (20)-(22) represents the motion of a perturbation of a circular orbit in the complex
plane.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com
221
For comparison purposes in our numerical illustration we use the methods mentioned above.
0
Method [a]
Method [b]
Err
-2
-4
-6
-8
-10
1
2
3
4
5
n
Fig. 3 Error Errmax for several values of n for the problem of Stiefel and Bettis [5].
The numerical results obtained for the two methods, with stepsizes equal to h =
analytical solution. F igure 3 shows the errors Errmax for several values of n.
1
2n ,
were compared with the
3.3 Two-Body Problem
The following system of coupled differential equations is considered. This system is well known as the Two-Body
problem:
y
y = − , y(0) = 1 − e, y (0) = 0,
r
z
z = − , z(0) = 0, z (0) = 1 − e2 ,
r
3
where r = (y 2 + z 2 ) and an analytical solution given by:
y(x) = cos(u) − e,
z(x) =
1 − e2 sin(u).
(23)
(24)
(25)
(26)
where u − e sin(u) − x = 0.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
222
G. Psihoyios and T.E. Simos: Symmetric Four-step Trigonometrically-fitted Methods
-2
Err
-4
-6
Method [a]
Method [b]
-8
-10
-12
1
2
3
n
Fig. 4 Error Errmax for several values of n for the Two-Body problem.
We solve the above problem for e = 0. For comparison purposes in our numerical illustration we use methods
[a] and [b] mentioned above.
The numerical results obtained for the two methods, with stepsizes equal to h = 21n , were compared with the
analytical solution. F igure 4 shows the errors Errmax for several values of n.
From the results of the above problems one can clearly see the efficiency of the new method.
References
[1]
[2]
[3]
[4]
[5]
J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Applic.
18, 189 (1976).
G.D. Quinlan and S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, The
Astronomical Journal 100, 1694 (1990).
G.D. Quinlan, Resonances and instabilities in symmetric multistep methods, submitted.
J.M. Franco and M. Palacios, High order P-stable multistep methods, Journal of Computational and Applied Mathematics 30, 1(1990).
E. Stiefel and D.G. Bettis, Stabilization of Cowells Method, Numerische Mathematik, 13, 154(1969).
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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