Evoltion of onentration by Eler method (Dirihlet) Evoltion of onentration by Eler method (Nemann).2 t n =.4n.2 t n =.4n : t n

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1 5 t = = (, y, z) t (, y, z, t) t = κ (68) κ [, ] (, ) = ( ) A ( /2)2 ep, A =., t =.. (69) 4πκt 4κt = /2 (, t) = for ( =, ) (Dirihlet ondition) (7) = for ( =, ) (Nemann ondition) (7) (68) (, t) = ( ) ( y)2 ep (y, )dy (72) 4πκt 4κt (69) = y = ((y, ) = δ(y ) δ() (, t) ( (, t) = ep ( ) 2 ) G(, t) (73) 4πκt 4κt N + i = i, i =,,, N t n = n t, n =,, (, t) ( i, t n ) = n i n+ i = n i + κ t ( n 2 i+ 2 n i + n ) i (74) 2

2 Evoltion of onentration by Eler method (Dirihlet) Evoltion of onentration by Eler method (Nemann).2 t n =.4n.2 t n =.4n : t n =.4n : t n =.4n [,] dq dt = Q(t) = (, t) 2 ( d = κ t 2 d = κ (, t)d (75) = = ) = (76) = t (, t) 5.2 Diffsion eqation in D by Simple Eler impliit real*8 (a-h,o-z) integer pot,gdraw,n,tma real*8 kappa parameter(n=5, kappa=., dt=., ibs=) dt parameter(tma=4, gdraw=4) dimension (:n),old(:n) *** d=.d/dfloat(n) d2i=.d/d/d pi=4.d*atan(.d) amp=.d- t_init=. *** 2

3 =d*(n/2) do i=,n =i*d (i)=amp/sqrt(2.*pi*t_init)*ep(-(-)*(-)/4/kappa/t_init) if((i).lt..d-2) (i)=.d *** if(ibs.eq.) then * write(6,*) Dirihlet Condition open(, file= diffsion_d.dat ) * ()=.d (n)=.d else write(6,*) Nemann Condition open(, file= diffsion_n.dat ) * ()=() (n)=(n-) endif *** Write (=/2,t) t=.d write(6,) format(/, Evoltion of (/2,t), /) write(6,) t,(n/2) do i=,n write(,) i*d, (i) write(,) do n=,tma *** * _old do i=,n old(i)=(i) * do i=,n- diff=kappa*(old(i+)-2.d*old(i)+old(i-))*d2i (i)=old(i)+diff*dt *** if(ibs.eq.) then ()=.d (n)=.d else ()=() (n)=(n-) endif * * *** if(mod(n,gdraw).eq.) then do i=,n write(,) i*d, (i) write(,) t=n*dt 22

4 write(6,) t,(n/2) endif format(, pe.3, 2, pe2.6) stop end gnplot gdiffsion.plt ips= ips=, ips= epsfile bondary= if(ips==) set terminal postsript eps pls olor "Times-Itali" 22 set nologsale set nologsale y set range [.:.] set yrange [:.5] set label "" set ylabel "" if(ips== && bondary==) set otpt "diffsion_d.eps" if(bondary==) set title "Evoltion of onentration by Eler method (Dirihlet)";\ plot diffsion_d.dat sing :2 title $t_n=.4n$ with lines ;\ :2-> y pase - "Dirihelt ondition" if(ips== && bondary==) set otpt "diffsion_n.eps" if(bondary==) set title "Evoltion of onentration by Eler method (Nemann)";\ plot diffsion_n.dat sing :2 title $t_n=.4n$ with lines ;\ pase - "Nemann ondition" 5.3. t = (, ) = 2. ) y t > U = t = ν 2, y >, t > (77) y2 (y, t) ν = µ/ρ ν =. (y =, t) = U, (y, t) y (78) (y, ) =. (79) Rayleigh y 23

5 .8 Rayleigh problem t n =.4n.2 Evoltion of onentration by Eler method (Dirihlet) t n =.4n.6.8 y : ν =. t n =.4n 2: t =.2, t n =.4n. y = U e (y (t), t) = U e y (t) = 4νt t t =.2 2 2κ 2, 6) t t + t (, t + t) = = 4π t ep t (8) ( ) ( y)2 (y, t)dy (8) 4κ t G( y, t)(y, t)dy (82) G(, t) (73) t y y + dy (y, t)dy t 24

6 t= = 2k t -D<<D 以内に Q の大部分が含まれているためには t=dt Dt Dt/4 t=2dt D y D/2 y D 3: y Q = t, 2 t [, ] 4: t [, ] [ /2, /2] t/4 Q(t) = = (, t + t)d = (83) G( y, t)d (84) 2, t = y y [G(, t) + 2G(, t) + G(, t)] (85) 2 [, ] 3 G y = (73) 2 4κ t > (86) 4 2 (86) 2 / (2κ t) ( /2, t/2) ( /2) 2 / 2κ( t/2) = 2 / 2κ2 t < (86) t/2 [ /2, /2] 4 25

7 = /5 =.2, 2 /(2κ) = 4 4 /2/. = 2 3 t =. t =.2 Crank-Niolson Sheme n n + n+ i = n i + κ t ( ) 2 2 n+ i+ 2n+ i + n+ i + n i+ 2 n i + n i n+ i n+ Impliit Sheme (87) (87) 2 + b 2 + b 2 + b b 2 + b n+ n+. n+ N n+ N = f n f n. f n N f n N (88) f n i = n i+ (2 b) n i + n i, b = 2 2 κ t. (89) (88) n+ i = n+ i = α i n+ i + β i (9) 2 + b α i n+ i+ + f n i + β i 2 + b α i (9) (9) α i, β i α i = Dirihlet = N = 2 + b α i, β i = f n i + β i 2 + b α i (92) n+ = α n+ + β = α =, β = (93) i = i = N (92) α i, β i n+ N = n+ N = α N n+ N + β N n+ i Gass (8) t =

8 Evoltion of onentration by Crank-Niolson (Dirihlet) dt=.4.2 t n =.4n : t =.4, t n =.4n. 6 t + =, (94) f() (, t) = f( t) ξ = t (, t) = f(ξ) f t = df ξ df = dξ t dξ, f = df ξ dξ = df dξ, t + = df ( + ) = (95) dξ ξ (, t) ξ = t f(ξ) = f( t) = (, t) t = (, ) = f() t > (, t) f() t (94) 6 6. n+ j n j t = n j+ n j 2 { for <.5 Type A (, ) = otherwise (96) (97) 27

9 t -t=ξ -2 -t=ξ - -t= t 2 t A B C D 6: t = ξ (, t) t = 7 > / j = j j, > (98) 8 ) < = j+ j, < (99) (94) n+ j n+ j n j t n j t n+ j n j t = n j+ n j 2 = n j n j, > () = n j+ n j, < () + n j+ 2n j + n j 2 2 (2) 2 9 () 28

10 .5 n+2 Flow diretion.5 n n t j- j j+ 7: Type A (96) 8: n + j n t 6.2. > = 2, 2. < (94) [, L] L L (, t)d = d = ((L, t) (, t)) (3) t (, t) = (L, t) (2) Q(t) t =, Q(t) = L (, t)d = (4) 6.3 (La-Wendroff) n+ j n j t = n j+ n j t ( ) 2 2 n j+ 2 n j + n j (5) 29

11 : Type A () 2: Type A () : Type B 22: Type B : Type B 3

12 2 t/2 2 / 2 2 t/2 2 Type A Type B 2, 22, 6.2 ( Type B (, ) = 2πσ ep ( ) 2 ) 2 σ 2, σ =.5 (6) 6.4 Convetive eqation in D by La-Wendroff or Central Differene *** Choie of the nmerial sheme swith=: Central differene swith=: La-Wendroff *** Choie of the initial ondition init_type=: Top hat init_type=: Gassian impliit real*8 (a-h,o-z) integer kint,kdraw,n,nt,tma,swith parameter(n=, tma=4, kint=2, kdraw=2) parameter(n=, tma=, kint=2, kdraw=2) parameter(=.d, dt=.2, width=.5, init_type=, swith=) dimension (:n),old(:n) harater*2 file(:2) *** Make parameters d=.d/dfloat(n) pi=4.d*atan(.d) sigma2=width*width amp=.d-/sqrt(2.*pi*sigma2) r=*dt/d file()= Central_differene file()= La-Wendroff if(swith==) open(,file= Central_differene.dat ) if(swith==) open(,file= La-Wendroff.dat ) 3

13 write(6,) file(swith),r format(,a2,, Corant Nmber=,pe.3,/) write(6,*) Conservation of Q *** Initial ondition and omptation of total Q(t)=\int (,t)d *** =d*(n/2) do i=,n =i*d (i)=.d if(init_type==.and. abs(-)<=width) (i)=.d if(init_type==) (i)=amp*ep(-.5d*(-)**2/sigma2) *** Otpt data for t= all otpt(,d,dt,n,,kint,kdraw) *** Time advaning by simple Eler do nt=,tma *** Copy to old do i=,n old(i)=(i) *** Choie of the nmerial sheme swith=: La-Wendroff swith=: Central differene do i=,n ipls=mod(i+,n) tmp=old(i)-.5d*r*(old(ipls)-old(i-)) diff=.5d*r*r*(old(ipls)-2.d*old(i)+old(i-)) (i)=tmp+swith*diff *** Enfore periodi bondary ondition ()=(n) *** Otpt data for t> all otpt(,d,dt,n,nt,kint,kdraw) stop end ******************************************************************** sbrotine otpt(,d,dt,n,nt,kint,kdraw) impliit real*8 (a-h,o-z) real*8 (:n) save Q_init if(mod(nt,kint).eq.) then Q=.d do i=,n Q=Q+(i) Q=Q*d if(nt==) Q_init=Q write(6,) nt*dt,q/q_init endif **** write Q 32

14 *** write (,t) if(mod(nt,kdraw).eq.) then do i=,n write(,2) i*d,(i) write(,2) endif format(,2(pe.3,)) 2 format(,2(pe2.5,)) retrn end (, t) t + = ν 2 2 (7) Brgers(948) : (, ) = sin ( > ) 33

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