1 u t = au (finite difference) u t = au Von Neumann
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- しじん なつ
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1 1 u t = au (finite difference) u t = au Von Neumann Von Neumann (Leap Frog) Leap Frog U t = au xx Neumann u t = cu x Neumann radiative boundary condition cyclic boundary condition) CFL condition u tt = c 2 u xx Staggard Grid
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3 u t = a + bu + cu x + du xx (a, b, c, d ) (1) u t x ut = a u t = bu u t = cu x u tt = c 2 u xx u xx + u yy = f(x, y) b b ( ) 1
4 1 u t = au 1.1 (finite difference) ( u) ( t) (grid, mesh ) u (t = t n, n = 1, 2, 3, ) du u(t + δt) u(t) (t) = lim dt δt 0 δt (t = t n) u du dt (t) u n+1 u n t u n t = t n u (1.1) du u(t) u(t δt) (t) = lim dt δt 0 δt du u(t + δt) u(t δt) (t) = lim dt δt 0 δt δt 0 (1.3) (1.4) (1.1) du/dt t=n t = (u n+1 u n )/ t du/dt t=n t = (u n+1 u n 1 )/2 t (1.5) du/dt t=n t = (u n u n 1 )/ t (1.5) t du/dt t Neumann (1.1) (1.2) (1.3) (1.4) 1.2 u t = au u t = au (1.6) 2
5 a u(t = 0) u(t) = exp (a t) u(t = 0) (1.7) (1.6) (1.6) (u n+1 u n )/ t = au n (1.8) u n n (t = t n) u n+1 u n+1 = tau n + u n (1.9) n = 1, 2, 3, 4, ( ) ASPEN EX DO LOOP SUBROUTINE u n U1 U2 100, 1000 u U1 U2 2 n = 1, 3, 5,... U1 U2 U2 U1 (n) : MAIN : U1 U2 U1 U2 U1 MOD FORTRAN n=1 ( ) U1 U2 U1 U2 DIMR( ) DIMI( ) REAL IMAG FORTRAN 100 DO LOOP GOTO 100 (THRESH) MAIN (SUBROUTINE ) CALL OPENPG CALL ( ) EX1 READY GO EX1 EX1 Input, Input, Input, Input, number of points time step real part of constant imaginary part of constant 3
6 t = n t n (NMAX) t (1.6) a (CASE ) t ( ) ( ) (CASE ) t Von Neumann Von Neumann u(n + 1) = λu(n) (1.10) λ λ λ < 1 λ = 1 λ > 1 ( ) ( ) ( ) λ 1 (1.9) (1.10) a iω ω a ω σ λu(n) = i tωu(n) + u(n) (1.11) λ = 1 + i tω (1.12) λ 2 = 1 + ( tω) 2 > 1 (1.13) 4
7 1.4 Von Neumann (u n+1 u n 1 )/(2 t) = iωu n (1.14) u n+1 u n+1 = 2i tωu n + u n 1 (1.15) u n+1 = λ 2 u n 1 u n = λu n 1 u n 1 λ 2 2i tωλ 1 = 0 (1.16) λ = i tω ± 1 ( tω) 2 (1.17) λ λ = ( tω) ( tω) 2 = 1 (1.18) tω 1 Leap Frog ( ) 1-2 λ 2 = 1 + ( tω) 2 t ( ) Leap Frog t 1/ω t tω 1 λ = tω + ( tω) 2 1 (1.19) ( tω > 1, ( tω) 2 1 > 0) (1.20) λ > 1 Leap Frog 1.5 (1.6) u n+1 = 2a tu n + u n 1 (1.21) n+1 n n-1 n=2 n=2 EX1 READY COPY NUMERCL2.FORT(EX1) NUMERCL2.FORT(EX2) EX1 EX2 5
8 EX.2 EX1 U1 U2 SUBROUTINE SUBROUTINE CENTER(UNXT, UNOW, CNST, DT) UNXT u n 1 UNOW u n UNXT u n+1 : MAIN : U1 U2 U1 U2 U1 U2 n=3 u n = 1 2 u EX1 U1,U2 EX2 EX2 EX1 CASE( ) 100 t ( CASE( ) ) (CASE ) 1000 t (CASE ) 100 t Neumann tω 1.6 (Leap Frog) u/ t = f(x, t) (u n+1 u n 1 )/(2 t) = f n (1.22) u n+1 = (2 t)f n + u n 1 (1.23) 6
9 Leap Frog ( ) n + 1 u n u n 1 leap frog (u t = au, a > 0) Neumann Ex1 Ex2 1.7 Leap Frog Leap Frog 3 (n + 1 n n 1 3 ) 2 ( ) u/ t = f(t) = 0 U( )= 1 U( )= 2 Neumann Leap Frog Neumann λ u n Λ n 1 n n 1 λ n 1 λ λ = i tω ± 1 ( tω) 2 t 0 λ λ 1, 1 (1.24) t 0 λ 1 λ EX3 Ex.3 EX1 EX2 EX2 EX3 EX3 1 2STEP 1 STEP 1 u 1 = /2 (1.25) u 2 = + /2 (1.26) U1 U2 7
10 WRITE(6,*) Input, difference of initial value READ(5,*) DIFFER U1 = DINIT - DIFFER/2 U2 = DINIT + DIFFER/2 DO 100 DO LOOP n=2 DIMR(2) DIMI(2) U2 DEFFER 0, 0.2, 0.5, 1, Ex ( ) 19 Leap Frog ( ) U n+1 U n 1 U n U n+1 = U n + t(du/dt) n + (1/2!) t 2 (d 2 U/dt 2 ) n + (1/3!) t 3 (d 3 U/dt 3 ) n + (1.27) U n 1 = U n t(du/dt) n + (1/2!) t 2 (d 2 U/dt 2 ) n (1/3!) t 3 (d 3 U/dt 3 ) n + (1.28) (du/dt) n n (t = t n) du/dt (U n+1 U n )/ t = (du/dt) n + (1/2!) t(d 2 U/dt 2 ) n + (1/3!) t 2 (d 3 U/dt 3 ) n (1.29) (U n+1 U n 1 )/(2 t) = (du/dt) n + (1/3!) t 2 (d 3 U/dt 3 ) n (1.30) (du/dt) n t 1 t > t 2 t t 2 1/ t 8
11 2 U t = au xx 2.1 U t = au xx (2.1) (2.1) Leap Frog) 2 U n (j) n j i j t x U t = au xx U(n + 1, j) = (2.2) : U t = au xx (2.3) 2.2 Neumann Neumann (2.2) (2.3) t, x t x x 10km 2 3km x t t t U n+1 (j) = U 0,n λ exp (ik xj) (2.4) U n (j) = U 0,n exp (ik xj) (2.5) U n+1 (j) = U 0,n λ n+1 exp (ik xj) (2.6) U n (j) = U 0,n λ n exp (ik xj) (2.7) 9
12 (2.6) (2.7) (2.2) (2.8) U 0 λ n e ik xj λ λ = (2.9) e ±ik x = cos (k x) ± i sin (k x) (2.10) t k x k > 0, x > 0 k x > 0 k x (two grid interval wave 2 x π/ x k x π λ < 1) t (2.11) Leap Frog (2.12) (2.15) (2.3) U 0 λ n e ik xj (2.12) U 0 λ n 1 e ik xj λ (2.13) e ±ik x = cos (k x) ± i sin (k x) (2.14) d = ta/ x 2 (2.15) Leap Frog 2.3 Neumann EX.4) SUBROUTINE FORWRD FORWRD SUBROUTINE FORWRD(DIMNXT, DIMNOW, ACOEF, DX, DT, JMAX) 10
13 JMAX DIMNOW U(j) at t = n t JMAX DIMNXT ACOEF (a) DX DT j=1, JMAX u x = 0 no-flux condition EX.4 MAIN PROGRAM DIM1 DIM2 n+1 DIM1 n DIM2 EX e-folding scale x = 0.1 =
14 3 u t = cu x 3.1 u t = cu x (c > 0) (3.1) (c > 0 c < 0 ) c > 0 t = (n + 1) t x = j x t = n t t = (n 1) t x = j x 3.5 x = j x x = (j 1) x 2 4 (3.1) U n+1 (j) = (3.2) (3.3) (3.4) (3.5) (3.2) (3.3) (3.4) (3.5) 3.2 Neumann (3.2) (3.5) Neumann (3.2) (3.5) Neumann Neumann c t/x µ 1 (3.2) (3.5)
15 Neumann 2 Neumann λ λ 2 + aλ + b = 0 a b λ = ( b ± b 2 4ac)/2a 2grid interval wave (k x = π) 3 Neumann µ 1 CFL condition µ = c t x 1 t x c 3.3 radiative boundary condition cyclic boundary condition) j=1 jmax u n+1 (JMAX) u n+1 (1) u n+1 (JMAX) u n (JMAX) u n (JMAX-1) J (1 JMAX) u n+1 (JMAX) u n+1 (1) u n (0) u = 0 u n+1 (1) = 0 radiative boundary condition u n+1 (JMAX) u n (JMAX-1) u n (JMAX+1) u n (JMAX+1) u n (JMAX+1) cyclic boundary condition u n+1 (JMAX) u n (JMAX+1) u n (1) u n+1 (1) u n (0) u n (JMAX) 1 13
16 3.4 EX.5 & 6 (3.2) (3.5) EX cyclic boundary condition 3.5 CFL condition U(n + 1, j) = U(n 1, j) tc(u(n, j + 1) U(n, j 1))/ x (3.6) (n+1,j) (n-1,j),(n,j+1),(n,j-1) t/ x x/ t x/ t c x/ t (3.7) CFL condition CFL condition CFL condition 3.6 EX.5 6 grid EX EX5A 6A EX5 6 DO 10 J = 1, JMAX 10 DIM1(J) = EXP(-((J-FJMID) / WIDTH)**2) * AMP 14
17 DO 10 J = 1, JMAX IF (ABS(J-FJMID).LT.WIDTH/2) THEN DIM1(J) = AMP ELSE DIM1(J) = 0 ENDIF 10 CONTINUE Grid 3.7 EX.5A EX.5A 6A U j / t = c(u j U j 1 )/ x (3.8) Uj U j 1 U j U t = c x {U j (U J x U j x + x2 2 U j 2 x 2 x3 6 U j t + c U j x 3 U j )} (3.9) x3 = c x 2 U j 2 x 2 c x2 3 U j 6 x 3 (3.10) Section 2 (3.10) c x 3.8 Neumann e ikx U j t + cu j+1 U j 1 = 0 (3.11) 2 x 15
18 U k U j (t) = Re{U k (t) exp (ik xj)} du k dt eik xj + c U k e ik x(j+1) U k e ik x(j 1) 2 x = 0 (3.12) exp (ik xj) exp (iθ) = cos (θ) + i sin (θ) du k dt + ikc sin (k x) U k = 0 (3.13) kδx U t + cu x = 0 U(t) = Re{U k (t) exp (ikx)} du k dt + ikcu k = 0 (3.14) (3.13) (3.13) sin (k x)/(k ) k x 0 π sin (k x)/ x 0 1 (k x = π) k x 0 sin (k x)/(k x) 1 c (dispersion) 3.9 EX.7 U t = V U x + ϵu xx Leap Frog cyclic boundary condition DX=1.0 C( )=0.1 ϵ( )=0.01 WORK(JMAX) MAIN PROGRAM SUBROUTINE SUBROUTINE ONESTP CALL ONESTP(DIM2, DIM1, WORK, C, EPS, DX, DT, JMAX) DIM1,DIM2 JMAX WORK SUBROUTINE 8 16
19 Neumann Neumann Neumann WORK n+1(j) = (3.15) EX6 u n UNOW u n+1 u n 1 UNXT EX6 u n+1 u n 1 u n+1 (j) n-1 u n 1 (j) UNXT(j) u n 1 (j) u n+1 (j) u n 1 (j) (3.15) u n+1 (j) n-1 u n 1 (j-1), u n 1 (j),u n 1 (j+1) u n+1 (j) u n 1 (j) u n+1 (j) u n+1 (j+1) u n+1 (j-1) SUBROUTINE u n+1 u n 1 u n+1 (j) UNXT(j) u n 1 u n+1 u n+1 (j+1) u n 1 (j) UNXT(j) u n+1 (j+1) u n+1 (j+1) u n+1, u n, u n 1 MAIN PROGRAM 3 MAIN DO 3 IF SUBROUTINE 3 MAIN 2 (3.15) u n u n 1 WORK SUBROUTINE u n 1 WORK u n 1 WORK SUBROUTINE WORK WORK SUBROUTINE MAIN WORK 17
20 4 u tt = c 2 u xx 4.1 u tt = c 2 u xx (4.1) c x (4.1) ( t + c x )( t c )u = 0 (4.2) x +c c η (4.1) u t = f(η) (4.3) η t = g(u) (4.4) f g (4.1) u tt = c 2 u xx u tt = (f(η)) t = f(η t ) = f(g(u)) (4.5) f g c 2 2 / x 2 f = g = c / x u t = cη x η t = cu x (4.6) (4.7) u η ( ) grid u η ( ) 18
21 (4.8) (4.9) Neumann CFL condition 4.3 n+1 n n-1 n=2 n=2 3.6 n=2 n=2 n=2 EX.8 (η = 0, u x = 0) η u 4.4 Staggard Grid EX.8 2 (4-1-10a,b)?? n+1 U n U η (n+1,j) U η (n,j-1) (n,j+1) (n,j) η grid 1/4 (staggered grid) j j j n n n EX.9 n,j u = 0 19
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9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
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m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
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2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
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Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx
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99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
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18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
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5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
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Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1 (2 ( BASIC BASIC download TUTORIAL.PDF http://hp.vector.co.jp/authors/va008683/
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[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin + 8 5 Clifford Spin 10 A 12 B 17 1 Clifford Spin D Euclid Clifford Γ µ, µ = 1,, D {Γ µ, Γ ν
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
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数値計算：常微分方程式
( ) 1 / 82 1 2 3 4 5 6 ( ) 2 / 82 ( ) 3 / 82 C θ l y m O x mg λ ( ) 4 / 82 θ t C J = ml 2 C mgl sin θ θ C J θ = mgl sin θ = θ ( ) 5 / 82 ω = θ J ω = mgl sin θ ω J = ml 2 θ = ω, ω = g l sin θ = θ ω ( )
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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数学の基礎訓練I
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