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1 (2) : 1

2 (1) dx dt = f(x,t) ( (t 0,t 1,...,t N ) ) h k = t k+1 t k. h k k h. x(t k ) x k. : 2

3 (2) :1. step. 1 : explicit( ) : ξ k+1 = ξ k +h k Ψ(t k,ξ k,h k ) implicit( ) : ξ k+1 = ξ k +h k Ψ(t k,t k+1,ξ k,ξ k+1,h k ) : 3

4 (3), Euler Ψ Runge-Kutta ( explicit ),.., x(t) dx dt = f(x,t). : 4

5 Runge-Kutta (1) Runge-Kutta 1., Φ(t k,x k,h k ), x k+1 = x k +h k Φ(t k,x k,h k ). Φ(t k,x k,h k ), Ψ(t,x(t),h k ). Ψ(t,x(t),h k ) Φ(t,x(t),h k ), τ(t,r). : 5

6 Runge-Kutta (2) C > 0, ρ > 0, r : 0 < r < ρ, t [a,b], τ(t,r) Cr p, p, p., x(t+r) x(t) x(t) x(t r),. : 6

7 Runge-Kutta (3) (h ), p, L > 0, k, x k ξ k el(b a) L Chp ([ ], pp ). : 7

8 Runge-Kutta (4) Runge-Kutta, f(x,t) Ψ p (p 1).. Heun Runge-Kutta. : 8

9 Runge-Kutta (5) Ψ f s, s (s-stage) Runge-Kutta ([ ]) s Runge-Kutta, p. : 9

10 Runge-Kutta (6) s p ([ ]), s = p., (, ). ξ k+1 = ξ k +h k Ψ(t k,ξ k,h k ) Ψ p. : 10

11 Runge-Kutta (7) 2 Runge-Kutta 2 Runge-Kutta, k 1 = f(ξ k,t k ), k 2 = f(ξ k +h k βk 1,t k +αh k ),Ψ(t k,ξ k ) = c 1 k 1 +c 2 k 2. α, β,c 1, c 2 2. : 11

12 Runge-Kutta (8) 2 explicit Runge-Kutta, α > 0, β = α, c 1 = 1 1, c 2α 2 = 1 2α, 2., 2 2 explicit Runge-Kutta, ([ ], pp ).. : 12

13 Runge-Kutta (9) 2 Runge-Kutta α β c 1 c 2 Heun Euler : 13

14 Runge-Kutta (10) Runge-Kutta, f Taylor,.,. Runge-Kutta,., Runge-Kutta,,,. : 14

15 Runge-Kutta (11) 4 Runge-Kutta 4 Runge-Kutta, k j = f(ξ k +h k l<j β jlk l,t k +α j h k ) ( j = 1,...,4, l<j β jl = α j ),Ψ(t k,ξ k ) = 4 j=1 c 4 jk j ( j=1 c j = 1). (j = 1 l<j ). : 15

16 Runge-Kutta (12) Runge 1/6 α 1 α 2 α 3 α 4 c 1 c 2 c 3 c β 21 = 1 2,β 32 = 1 2,β 43 = 1, RK4, RK41, Runge-Kutta, : 16

17 Runge-Kutta (13) Kutta 1/8 α 1 α 2 α 3 α 4 c 1 c 2 c 3 c β 21 = 1 3,β 31 = 1 3,β 32 = 1 β 41 = 1,β 42 = 1,β 43 = :

18 Runge-Kutta (14), Runge-Kutta. s Ψ(t k,ξ k ) = c j k j, s j=1 c j = 1 j=1. k j. j = 1,...,s. : 18

19 Runge-Kutta (15) Explicit( ) : l<j β jl = α j, k j = f(ξ k +h k β jl k l,t k +α j h k ), l<j Implicit( ) : s l=1 β jl = α j, s k j = f(ξ k +h k β jl k l,t k +α j h k ) l=1 : 19

20 Runge-Kutta (16) Euler 1 Runge-Kutta. s Runge-Kutta, s α 1,...,α s, s c 1,...,c s s 2 (β il )., α j = s l=1 β jl. :

21 Runge-Kutta (17) Runge-Kutta, Butcher.,. α 1 β 11 β 12 β 1s α 2 β 21 β 22 β 2s.... α s β s1 β s2 β ss c 1 c 2 c s : 21

22 Runge-Kutta (18) explicit Butcher,. s = p = 2, s = p = 3, s = p = 4, Runge-Kutta ( Butcher ) ([Butcher]). : 22

23 Runge-Kutta (19) s = p = 2 explicit. s = p = 3, s = p = 4,, ([Butcher] )., Butcher ([ ]). : 23

24 1 (s = p = 1) [ ] Euler Euler (s = p = 2) [ ] Heun Euler Crank Nicolson : 24

25 3 (s = p = 3) [ ] Heun Kutta : 25

26 4 (s = p = 4) [ ], [Hairer] Vol. 1 Runge 1/6 Kutta 1/ : 26

27 Runge-Kutta (23) B = (β jl ) 1 j s,1 l s Butcher, α = (α 1,...,α s ) T, c = (c 1,...,c s ), Butcher (B,α,c). : 27

28 Runge-Kutta (24) explicit Runge-Kutta, p 5 p p, p 7 p+1 p, p 8 p+2 p ([Hairer et al], Vol. 1). s. : 28

29 Runge-Kutta (25) 5 6, 6 7, 8 11, ([Hairer et al], Vol. 1).,. : 29

30 (1),., Butcher (B,α,c) p+1,butcher (B,α,c) p,. : 30

31 (2) (p,p+1), Butcher., s p+1 p ([ ]). : 31

32 (3) Runge-Kutta Runge-Kutta ([ ]).,. (B,α,c) Runge-Kutta (ξ k ),(B,α,c) Runge- Kutta (ξ k ). : 32

33 (4), ξ k ξ k, (,, ).. : 33

34 (5),, ( ). [ ]. Scilab RKF45 Runge-Kutta, : 34

35 implicit Runge-Kutta, implicit Runge-Kutta ( ) differential-algebraic equation.,,. implicit Runge-Kutta [Hairer et el.] [Butcher]. : 35

36 d dt x = Ax, A,. A λ m, λ M. Reλ M / Reλ m... : 36

37 (1), ξ k+1, ξ k, (ξ k L,...,ξ k ) ( ). Runge Kutta,,, (ξ k L,...,ξ k ). : 37

38 (2), ξ k+l (ξ k,...,ξ k+l 1 )., h ( [Hairer et al.], Vol. 1 ). : 38

39 (3) Runge Kutta, f(ξ k,t k ),..., f(ξ k+l,t k+l ) ( : L ) ξ k+l = ξ k +h l=0 β(0) l f(ξ k+l,t k+l ) β (0) l (0 l L). ξ k : 39

40 (4) ξ k+l ξ k = h ( L ) l=0 β(0) l f(ξ k+l,t k+l ) {ξ k+l } 0 l L, ( ). ( L L ) α l ξ k+l = h β l f(ξ k+l,t k+l ) l=0 l=0 : 40

41 (5) {α l } 0 l L, {β l } 0 l L,., ξ k+l implicit( ), ξ k+l explicit( ). 1. : 41

42 (6) ( ), ([ ]),. L l=0 α lξ k+l = hψ ( t k,...,t k+l,ξ k,...,ξ k+l ;h ),,. : 42

43 (7) L, ξ 0,..., ξ L 1 (1 ). ( ) {α l } 0 l L {β l } 0 l L.,., ([ ]): (1) α 0 +α 1 z + +α L z L 1, 1, (2) L l=0 α l = 0, (3) l l=0 lα l L l=0 β l = 0. : 43

44 , explicit, implicit,., explicit, implicit. explicit implicit. : 44

45 Adams (1) dx = f(x,t), x(t dt k+l 1 ) = x k+l 1. x k+l x k+l 1 = t k+l t k+l 1 f(ϕ(τ,t k,x k ),τ)dτ, ϕ(τ,t k,x k ). : 45

46 Adams (2), ϕ(τ,t k,x k ) τ 1, Lagrange. (t k,x k ),...,(t k+l 1,x k+l 1 ), x k+l x k+l 1 L 1 l=0 β lf(x k+l,t l ),(t k,x k ),...,(t k+l,x k+l ),x k+l x k+l 1 L l=0 β lf(x k+l,t l ). : 46

47 Adams (3) x j ξ j (k j k +L),., Adams. (ξ k+l,t k+l ) Adams Bashforth, Adams Moulton. : 47

48 Adams (4) Adams Bashforth explicit, Adams Moulton implicit. {β l } 0 l L, Lagrange. L,,. : 48

49 Adams (5) t t k+1, L=2 Adams Bashforth. t k f(ξ k,t k ), t k+1 f(ξ k+1,t k +1) Lagrange, f(ξ t k t k,t k ) + t t k f(ξ k+1 t k+1 t k+1,t k+1 ) k tk+2, t k+1 t k = h, = t k t k+1 1 2h (t t k+1) 2 t k+2 = h tk+2 t k+1 2, t t k = 1 t k+1 t k+1 t k 2h (t t k) 2 t k+2 = 3h t k+1 2., L=2 Adams Bashforth, ( ξ k+2 ξ k+1 = h 1 2 f(ξ k,t k )+ 3 ) 2 f(ξ k+1,t k+1 ). t k+1 t t k+1 : 49

50 BDF (1) BDF Backward Difference Formula. implicit. BDF ([ ]). ξ k+l (ξ k,...,ξ k+l 1 ). : 50

51 BDF (2) ξ k+l, (t k,ξ k ), (t k+1,ξ k+1 ),..., (t k+l,ξ k+l ) L Lagrange. p(t).,t k+l 1 = t k+l h, t k+l 2 = t k+l 2h,.... : 51

52 BDF (3) p(t) ξ k+l.,p(t) t = t k+l dx = f(x,t) dt. : 52

53 BDF (4) d dt p(t) = f(ξ k+l,t) ξ k+l t=tk+l BDF. Lagrange, t = t k+l, ( ). : 53

54 BDF (5) L BDF L = 1 : ξ k +ξ k+1 = hf(ξ k+1,t k+1 ) 1 L = 2 : ξ 2 k 2ξ k+1 + 3ξ 2 k+2 = hf(ξ k+2,t k+2 ) L = 3 : 1ξ 3 k + 3ξ 2 k+1 3ξ k ξ 6 k+3 = hf(ξ k+3,t k+3 ) BDF L > 6. : 54

55 Scilab (1) ( ) ( )( d x = dt 1 0 x 2 x 1 x 2 ), ( ) ( ) x 1 (0) 1 = x 2 (0) 0. x 1 (t) = cost, x 2 (t) = sint., BDF,. : 55

56 // ( ) function dx=f(t,x) dx=[0-1;1 0]*x; endfunction t0=0; //. t=0:.1:100; x0=[1;0]; : 56

57 ode. y=ode(x0,t0,t,f); // y=ode("adams",x0,t0,t,f); //Adams y=ode("stiff",x0,t0,t,f); //BDF y=ode("rk",x0,t0,t,f); //Runge-Kutta y=ode("rkf",x0,t0,t,f); //RKF45 Adams BDF : 57

58 ( y(t)( ) Frobenius ) Adams BDF Runge-Kutta RKF : 58

59 Scilab (5) [ ]. ( ) ( )( ) ( ) d dt x 1 x 2 = x 1 x 2 + cost 1999cost sint x 1 (0) = 1, x 2 (0) = 2. x 1 (t) = e t, x 2 (t) = e t +cost ( Mathematica )., : 59

60 // ( ) function dx=f(t,x) dx=[-2 1; ]*x+.. [-cos(t);1999*cos(t)-sin(t)]; endfunction // :.. t0=0; t=0:.5:100; x0=[1;2]; : 60

61 . y=ode(x0,t0,t,f); // y=ode("adams",x0,t0,t,f); //Adams y=ode("stiff",x0,t0,t,f); //BDF y=ode("rk",x0,t0,t,f); //Runge-Kutta y=ode("rkf",x0,t0,t,f); //RKF45 : 61

62 ( y(t)( ) Frobenius ) Adams BDF Runge-Kutta RKF : 62

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x =

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x = 3 MATLAB Runge-Kutta Butcher 3. Taylor Taylor y(x 0 + h) = y(x 0 ) + h y (x 0 ) + h! y (x 0 ) + Taylor 3. Euler, Runge-Kutta Adams Implicit Euler, Implicit Runge-Kutta Gear y n+ y n (n+ ) y n+ y n+ y n+