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1

2

3 d 2 x i dt 2 = j i Gm j x j x i x j x i 3, (1) 2 10

4 2 O(N 2 ) N

5 ( )

6

7 k 1 = x i + h 2 f(x i, t i ) x i+1 = x i + hf(k 1, t i + h/2) (2) y step 2 y i step 1 h x i x x i+1 2 ( 2 )

8 x n+1 k i = x n + h s i=1 = f(x n + h s b i k i j=1 a ij k j, t n + c i h) (3) s : number of stages a ij, b i, c i a c c i = s j=1 a ij (4)

9 2 s = 2 x n+1 = x n + h(k 1 b 1 + k 2 b 2 ) k 1 = f(x n + h(a 11 k 1 + a 12 k 2 ), t n + c 1 h) k 2 = f(x n + h(a 21 k 1 + a 22 k 2 ), t n + c 2 h) (5) a ij (j i) 0 k 1 a ij (j > i) 0 k i k i (semi-implicit) k 1 k 2

10 k i

11 Runge-Kutta x n+1 = x n + h(k 1 /6 + k 2 /3 + k 3 /3 + k 4 /6) k 1 = f(x n, t n ) k 2 k 3 = f(x n + hk 1 /2, t n + h/2) = f(x n + hk 2 /2, t n + h/2) k 4 = f(x n + hk 3, t n + h) (6)

12 Runge-Kutta 1. a ij i j =

13 s s 10 p p s 2 2 2

14 Dormand Prince (Fotran C )

15 : : (variable step) / (adaptive step) h =

16 RK x n+1 = x n + s i=1 b i k i (7) k i b i b i x = x n + s i=1 b i k i (8) x embedded Runge-Kutta-Fehlberg Dormand-Prince

17 ( ) s 2s 8 2s

18 : t i x i t i+1 x i+1 : t i

19 : f f i-3 i-2 i-1 i i+1 t

20 i i + 1 i p i p P (t j ) = f j = f(x j, t j ) (i p j i) (9) i + 1 x i+1 x i+1 = x i + t i+1 t i P (t)dt (10) h p x i+1 = x i + h p l=0 a pj f i l (11)

21 2 p = 1 P (t) = f i f i 1 f i (t t i ) (12) h x i+1 = x i + h 2 (3f i f i 1 ) (13)

22 t i+1 h p x i+1 = x i + h p 1 l= 1 b pj f i l (14) p = 1 x i+1 = x i + h 2 (f i + f i+1 ) (15)

23

24 - - (PEC) - (PE) P(EC) 2 PE(CE) 2

25 t 0 x 0 : : :

26 : :

27 2

28 ( )

29 2 2 : 2

30

31 : : ( )

32

33 : ρ r α σ r (2 α)/2 (16) : 1/ Gρ r α/2

34 2 0 2 : : 0

35 : : O(N) 2 : O(N 1/3 )

36 ( ) 2

37 t i t i 1. t i + t i i t i ti : t i + t i n Time

38 PEC

39 t = η a 1 a (2) 1 + ȧ 1 2 ȧ 1 a (3) 1 + a (2) (17) 1 2

40 :

41 3 :? /

42 : 2 leap frog leapfrog Verlet 2nd order ABM 2nd order Stömer-Cowell 4

43 leap frog v i+1/2 = v i 1/2 + ta(x i ) (18) x i+1 = x i + tv i+1/2 (19) ±1/2 v 1/2 = v + ta(x 0 )/2 (20) v i = v i 1/2 + ta(x i )/2 (21)

44 x i+1 = x i + tv i + t 2 a(x i )/2 (22) v i+1 = v i + t[a(x i ) + a(x i+1 )]/2 (23) PC v i+1/2 = v i + ta(x i )/2 (24) x i+1 = x i + tv i+1/2 (25) v i+1 = v i+1/2 + ta(x i+1 )/2 (26) x i 1, x i, x i+1

45 leap frog 2 :

46 d 2 x = x (27) dt2 2 (x, v) = (1, 0) 1/4

47

48 H = 1 2 (x2 + v 2 ) h2 4 x2 (28) h 2 ( 1 )

49 d 2 x dt = 2 x3 (29) 2 (x, v) = (1, 0) 1/8

50

51 1. 2.

52

53 1. 2.

54 ? : :

55 : T (p) + V (q)

56

57 Ruth 4 S 4 (h) = L(d 1 h)l(d 2 h)l(d 1 h), d 1 = 1/(2 2 1/3 ), d 2 = 1 2d 1 = 2 1/3 /(2 2 1/3 ) (30) : L(h) h 1

58 4 h t t i i+1 2 1/3 2 ( ) ( )

59 : d 1 = d 7 = d 2 = d 6 = d 3 = d 5 = d 4 = (31)

60 ? H p H H H H = H + h p+1 H p + O(h p+2 ) (32) H ( ) ( )

61 ( ) H H = H 1 + H 2 + (33)

62 1 : ( ) : Dormand

63

64 MVS + ( ) : H = T (p) + V (q) (34) p p p t H q (35) q

65 : H = T (p) + V 1 (q) + V 2 (q) (36) V 1 V 2 H 1 = T (p) + V 1 (q) (37) V 2 p p t V 2 (38) q 2

66 : x x + tv MVS: V 1 KS

67 ( ) dy dx = f(x, y) (39) y i+1 = F (x i, y i, f, x) (40) y i = F (x i+1, y i+1, f, x) (41) 1

68 y i+1 = y i + h 2 (f i + f i+1 ) (42)

69

70 :

71 RK

72 = 2 : α 0 x i+p + α p x i = h 2 (β 0 f i+p + β p f i ) (43) β 0 = 0 α i = α p i, β i = β p i (44) x i+1 2x i + x i 1 = h 2 f i (45)

73 (x i,...x i+p 1 ) x i+p (x i+p,...x i+1 ) x i

74 Quinlan Tremaine 14 Quinlan & Tremaine

75 : 6 6

76 : 4 x 1 = x 0 + t 2 (v 1 + v 0 ) t2 12 (a 1 a 0 ), (46) v 1 = v 0 + t 2 (a 1 + a 0 ) t2 12 (ȧ 1 ȧ 0 ). (47) ( )

77 ( )

78 e = 0.9

79 t 0 t 0 t 1 t 1 t = 2 1[h(ξ 0) + h(ξ 1 )] (48) h(ξ) ξ 1 t

80

81

82 Cano Sanz-Serna (a), (b) (c) 1

83

A

A A05-132 2010 2 11 1 1 3 1.1.......................................... 3 1.2..................................... 3 1.3..................................... 3 2 4 2.1............................... 4 2.2

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