A STUDY OF GENERALIZED ADAMS-MOULTON METHOD FOR THE SATELLITE ORBIT DETERMINATION PROBLEM

Transcription

1 Korean J Math 2 (23), No 3, pp A STUDY OF GENERALIZED ADAMS-MOULTON METHOD FOR THE SATELLITE ORBIT DETERMINATION PROBLEM Bum Il Hong and Nahmwoo Hahm Abstract In this paper, a generalized Adams-Moulton method that is a m-step method derived by using the Taylor s series is proposed to solve the satellite orbit determination problem We show that our proposed method has produced much smaller error than the original Adams-Moulton method Finally, the accuracy performance is demonstrated in the satellite orbit correction problem by giving a numerical example Introduction Because a strongly stable multistep method in terms of round-off errors produces a relatively accurate approximation solution, the Adams- Bashforth-Moulton predictor-corrector method that is strongly stable is effectively used in many packages [, 4] In addition, since the multistep integrator needs relatively small number of function values [5], a multistep method for the orbit prediction and correction determination problem is preferable to a single step method In [3], Hahm and Hong suggested a generalized Adams-Bashforth method that is explicit for the Received June, 23 Revised August 5, 23 Accepted August 7, 23 2 Mathematics Subject Classification: 65L99, 65L6, 65Z5 Key words and phrases: Adams-Moulton method, multistep, orbit prediction problem Corresponding author This research was supported by Kyung Hee University Research Fund, 22 (KHU-2333) c The Kangwon-Kyungki Mathematical Society, 23 This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License ( -nc/3/) which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited

2 272 Bum Il Hong and Nahmwoo Hahm satellite orbit determination problem In this paper, we propose a generalized Adams-Moulton method for the same problem and our method is implicit Note that the original Adams-Moulton method is of the form () y i+ = y i + h b l f(t i l, y i l ) where b, b,, b are constants to be determined In this paper, by utilizing the Taylor s series, we formulate the generalized Adams- Moulton method associated with error control parameters (2) y i+ = k= a k y i k + h b l f(t i l, y i l ) where {a, a,, a } and {b, b,, b } are constants to be determined Then this method becomes a special case of the general multistep method and is only the method that can produce smaller local truncation error than the original Adams-Moulton method even if there are infinitely many strongly stable multistep methods For the comparison purposes, the two body problem of the Earth s satellite is integrated numerically The two body problem is considered since it has an exact solution that can be used in the error quantification The satellite for the numerical integration is a Geoscience Laser Altimeter System type low-altitude satellite [2] 2 Preliminaries Let the first-order initial-value problem be of the form (2) y = f(t, y); a t b, y(a) = y Suppose that h = b a, t N i = a + ih for i =,,, N, and that y i is the approximation to y(t i ) for each i =,,, Then the original m-step Adams-Moulton (AM) method is represented by (22) y i+ = y i + h b l f(t i l, y i l )

3 A Generalization of the Adams-Moulton Method 273 for i = m, m,, N, where b, b,, b are constants satisfying the property (23) = Moreover, this method is implicit since y i+ occurs on both sides of (22) Note that the Taylor s series for y i+ is given by (24) y i+ = y i + D i,j h j where (25) D i,j = j! j= b l d j y dt j (t i) In other word, D i,j is an operator representing the j-th derivative of y at t i divided by j! Therefore, we can easily show that the Taylor s series for y i k is (26) y i k = y i + ( k) j D i,j h j j= since t i k = t i kh As a result, if we use the Taylor s series of y i k (26) for i =, m,, N, then we have the implicit m-step generalized Adams-Moulton (GAM) method for solving the problem (2) given by the difference equation (27) y i+ = k= a k y i k + h b l f(t i l, y i l ) where {a, a,, a } and {b, b,, b } are constants to be determined Remark 2 In practice, each a k should be constrained to provide the method with the roundoff stability It is well known that the roots of the characteristic equation (28) λ m a λ a λ m 2 a = must satisfy the root conditions to be a strongly stable method: Criterion λ = is a simple root and is the only root of magnitude one

4 274 Bum Il Hong and Nahmwoo Hahm Criterion 2 All roots except λ = have absolute value less than For all roots λ, λ < except λ = Remark 22 If we set a = and a = = a = in the GAM method (27), then we simply obtain the original AM method (22) 3 Main results In this subsection, the generalized Adams-Moulton method that is a multistep method is derived by utilizing the Taylor s series The coefficient matrix and the error vector of the generalized Adams-Moulton method are formulated Strongly stable multistep methods can be obtained by choosing appropriate values of parameters associated with the local truncation error The formula for the local truncation error gives an idea how to choose such values, however, it might not meet with good results because of the accumulative errors It is known that those parameters should be non-negative 3 The local truncation error of the GAM Method In this subsection, we actually compute the local truncation error τ i+ (h) at each step of the m-step GAM method which is smaller than that of the original m-step AM method Theorem 3 For i =, m,, N, the local truncation error τ i+ (h) for the GAM method (27) is (3) { } τ i+ (h) = ( k) m+2 a k (m + 2)( l) m+ b l D i,m+2 h m+ k= Remark 32 If we apply Criterion to (28), then we can obtain a by the equation (32) a = During the proof of Theorem 3, we will show that each b l can be expressed as a linear combination of {a, a,, a } However, due to (32), it is sufficient to determine {a,, a } only k= a k

5 A Generalization of the Adams-Moulton Method 275 Proof To compute the local truncation error τ i+ (h), let us apply the Taylor s series expansion to the equation f = y by considering that f is a function of t Then the equation f = y is in the following form (33) f(t i l, y i l ) = j( l) j D i,j h j j= Substituting (24), (26) and (33) into (27) and equating coefficients of the same powers of h give the system of equations, (34) = ( k) j a k + k= j( l) j b l for j =, 2,, m + Therefore, a simple calculation gives that the local truncation error τ i+ (h) at this step is (35) τ i+ (h) = ( y i+ k= ) a k y i k /h l= b l f(t i l, y i l ) { } = ( k) m+2 a k (m + 2)( l) m+ b l D i,m+2 h m+ k= Remark 33 From the equation (34), we have two things mentioned earlier One is that, if we put j = in the equation (34), then (36) = b l k= ka k Therefore, since a = and a = = a = in the AM method, we have = as shown in the equation (23) The other is that each b l in the equation (34) is expressed in the linear combination of {a,, a } only as mentioned in Remark 32 b l

6 276 Bum Il Hong and Nahmwoo Hahm 32 The vector and matrix form of the GAM Method In this subsection, we use vectors and matrices to represent the GAM method for a convenient computer computation That is, we theoretically compute the coefficient matrix C and the error vector ẽ of the GAM method First, we rewrite the system of equations in (34) as a vector-matrix form For the first sum (37) ( k) j a k for j =, 2,, m + k= of the equation (34), we let A be an (m + ) m coefficient matrix of (37) such that (k+)-th (38) A = ( k) j j-th for j =, 2,, m + and k =,,, m Note that the first column of A is associated with the constant a in (37) Therefore, we can easily see that the first column of A becomes a zero vector because ( k) j a k = when k = Similarly, we define an (m + ) (m + ) coefficient matrix B of the second sum (39) j( l) j b l for j =, 2,, m + by (l+2)-th (3) B = j( l) j j-th for j =, 2,, m + and l =,,, m But only in the case of j = and l =, we set B 2 =

7 A Generalization of the Adams-Moulton Method 277 Also, let ã be an m-dimensional column vector and let b and be (m + )-dimensional vectors such that b b a (3) ã =, b = b and = a b Then the system of equations (34) is reduced to a vector-matrix form (32) = Aã + Bb From the equation (32), we end up with (33) Bb = + Aã We now define a new matrix à by replacing the first column of the matrix A that is zero vector by a column vector Then the new matrix à is of the form [ ] (34) à = A j,k+ for j =, 2,, m + ; k =,, m As we can see that the matrix à is the same as the matrix A except for the first column Therefore we have (35) Ãã = + Aã From (33) and (35), we have (36) Bb = Ãã Let (37) C = B à Then, by (36) and (37), b can be expressed in a simple form (38) b = Cã So, equation (38) shows that b l for each l =, 2,, m + is a linear combination of {a,, a } If we set (39) b = B and C = B A,

8 278 Bum Il Hong and Nahmwoo Hahm then, similar to Ã, the matrix C is the same as the matrix C except for the first column In fact, the first column of C is b so that C can be expressed as ] Cj,k+ (32) C = [b for j =, 2,, m + ; k =,, m Now, combining (35), (38) and (39) gives that b can be expressed in an additional form (32) b = b + Cã Note that since the AM method has (322) ã T = [ ], equation (32) becomes b = b as expected and T b = T B = b l = as shown in (23) Moreover, if we compare (36) to (32), an easy computation shows that (323) T C = [ m ] 33 Error Analysis In this paper, we theoretically find ã that provides a smaller local truncation error τ i+ (h) in (35)of the GAM method than that of the AM method For simplicity, let define ɛ by (324) ɛ = ( k) m+2 a k (m + 2)( l) m+ b l k= Then local truncation error τ i+ (h) in (35) is (325) τ i+ (h) = ɛd i,m+2 h m+ We now express ɛ in the equation (324) in terms of vectors and matrices If we define an m-dimensional column vector c and an m + - dimensional column vector d by (326) c = ( k) m+2 (k + )-th and d = (m + 2)( l) m+ (l + 2)-th

9 A Generalization of the Adams-Moulton Method 279 for k =,,, m and l =,,, m, then by (32) and (326), ɛ in (324) becomes (327) (328) (329) ɛ = + d T b + c T ã = + d T b + (d T C + c T )ã = ɛ o + e T ã where ɛ o = + d T b and e T = d T C + c T Note that the first entry of e is zero Therefore if we replace the first entry of e by ɛ o, we get a new column vector ẽ such that ] (33) ẽ T = [ɛ o e Tk+ for k =, 2,, m As a result, this makes us have a compact form of ɛ (33) ɛ = ẽ T ã Since e T ã = in the AM method, it is easy to see that the local truncation error of the AM method is as follows, (332) ɛ = ɛ o and τ i+ (h) = ɛ o D i,m+2 h m+ 34 Numerical values of coefficient matrix C and the error vector ẽ In this subsection, we provide numerical values of the coefficient matrix C and the error vector ẽ of the GAM method through the C program for the numerical computation As we can see in Subsections 32 and 33, the first column of C and the first entry of ẽ represent the AM method The followings are numerical values of the coefficient matrix C and the error vector ẽ of the GAM method Because the error has been accumulated during the numerical computation, all numerical values of ẽ below, except the first entry, are not non-negative always But 2, 4 and 6-step methods do have non-negative values except the first entry 2-step method: (333) C = and ẽ T /4! = [ ]

10 28 Bum Il Hong and Nahmwoo Hahm 3-step method: 9 (334) C = and ẽt /5! = [ ] step method: (335) C = , and ẽ T /6! = [ ] 27 5-step method: C = , (336) and ẽ T /7! = [ ] 52 6-step method: (337) C = 648 ẽ T /8! = , [ ]

11 4 Numerical result A Generalization of the Adams-Moulton Method 28 Since the two body problem of the Earth s satellite has an exact solution, we choose it as a numerical example for the error quantification of the GAM method Also, this numerical integration is a Geoscience Laser Altimeter System type low-altitude satellite [2] We consider the case that our satellite problem has a low altitude about 8 km The equation of a satellite orbit prediction problem is given by (4) [ṙ ] [ = v v (µ/r 3 )r where r and v are position and velocity vectors, respectively; µ is a gravitational constant and r is the magnitude of r with the initial condition (42) r = and v = In practice, low-step methods give relatively low accuracy, we apply the 6-step method to this example As shown in (337), all entries of ẽ are non-negative except for the first element Therefore, the error of the GAM method should be less than or equal to the error of the AM method in each case of a (43) ã =, ã 2 = a 2, ã 3 = a 3 ], ã 4 = a 4, ã 5 = for a k ; k =, 2,, 5 Figure shows the results of these when the a k is increased by from to The error is represented by the positional root mean squares (rms) As the a k approaches to, it becomes unstable because of the Criterion 2 Since the unstable method is not necessary, unstable cases are omitted in the figure In fact, ã 5 reduces the error significantly Since the error has been accumulated by its nature, this contradictable behavior can be explained in a way that the approximate solution y i 5 contains less error than y i 4, y i 3,, y i For the same reason, it is shown that ã 5 reduces the error much less than ã, ã 2, ã 3 and ã 4 a 5

12 282 Bum Il Hong and Nahmwoo Hahm 2 GAM Method for Satellite Orbit Prediction Problem AM method a a 2 a 4 a 3 rms (mm) a 4 (a 5 =9) a a k Figure GAM Method for Satellite Orbit Prediction Problem Among all cases of (43), the minimum error occurs when a 5 = 9 in ã 5 Moreover, we tried the following cases to find the better approximation: ã T,5 = [ a 9 ], ã T 2,5 = [ a 2 9 ], ã T 3,5 = [ a3 9 ] and ã T 4,5 = [ a 4 9 ], and found that ã i,5 for i =, 2, 3 are very unstable while ã 4,5 provides a smaller error as shown as the lowest curve in the figure Consequently, a better empirical values for ã is Better 6-step GAM method (empirical): ã T = [ 9 9 ] Acknowledgements The authors wish to express their gratitude to Prof S Lim of University of New South Wales for offering a numerical example of the satellite orbit determination problem and giving valuable comments The authors also wish to thank the reviewer for his/her valuable comments and suggestions

13 A Generalization of the Adams-Moulton Method 283 References [] R L Burden and J D Faires, Numerical Analysis, PWS Publishing Company, Boston, 993 [2] GLAS DESIGN TEAM, EOS ALT/GLAS Phase-A Study: Geoscience Laser Altimeter System, NASA Goddard Space Flight Center, Greenbelt, Maryland, 993 [3] N Hahm and B I Hong, A Generalization of the Adam-Bashforth Method, Honam Mathematical J, 32 (2), [4] D Kincaid and W Cheney, Numerical Analysis, Brooks/Cole Publishing Company, Pacific Grove, 99 [5] D D Rowlands, J J MaCarthy, M H Torrence and R G Williamson, Multi- Rate Numerical Integration of Satellite Orbits for Increased Computational Efficiency, Journal of the Astronautical Sciences, 43 (995), 89 Department of Applied Mathematics Kyung Hee University Yongin 446-7, Korea bihong@khuackr Department of Mathematics Incheon National University Incheon , Korea nhahm@incheonackr