差分法入門テキスト（第１講）
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4 Edmund Gunter W.Schickrd Blise Pscl Isc Newton Gottfried W. Leibniz Chrles Bbbge IBM Jcgurd Howrd H. Aiken IBM 944 Mrk H. Hollerith IBM 943 J. Mnchly & J. P. Eckert Aberdeen ENI AC (Electronic Numericl Integrtor nd Clcultor ) IBM 60, 60, 650 UNIVAC EDVAC FORTRAN IBM TOSBAC 500 4
5 Apple II 8k 980 ALGOL Algorithmic Lnguge BASIC COBOL PL/I 950 $0, $ MFLOPS FLOPS M GFLOPS G , TFLOPS T IBM IBM CRAY-YMP
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7 ( f x) x x x f '( x) 0 f ( x + h) f ( x ) = (3.) h h 0 0 f '( x0 ) lim 0 h f ( x0 + h) f ( x0) f '( x0) = (3.) h h B A (3.) (3.) y y= f() x B A h x 0 x0 + h x h x = x 0 x = x 0
8 x = x h x= x 0 f ( x0) f ( x0 h) f '( x0) = h f ( x0+ h) f ( x0 h) f '( x0) = h f( x0 + h) x= x 0 3 h h f( x0 + h) = f( x0) + f '( x0) h+ f ''( x0) + f '''( x0) +...! 3! f '( x ) 0 f( x0 + h) f( x0) h h f '( x0) = f ''( x0) f '''( x0) +... h! 3! f( x0 + h) f( x0) = + Oh ( ) h Oh ( ) h Oh ( ) f( x0 h) x= x 0 3 h h f( x0 h) = f( x0) f '( x0) h+ f ''( x0) f '''( x0) +...! 3! h Oh ( ) 0
9 3 h f '''( x0) f ( x0 + h) f ( x0 h) = hf '( x0) ! f ( x + h) f ( x h) '( ) = + ( ) h 0 0 f x0 Oh h f '( x0 + h/) f '( x0 h/) f ''( x0) = h f( x0 + h) f( x0) f( x0) f( x0 h) = h h h f( x0 + h) f ( x0) + f( x0 h) = h h f
10 () () k (3) c V = x y z ( xyz,, ) z f x x f y y (x, y, z) z f y y+ y f x x+ x x y
11 ρ m= ρ V= ρ x y z (4.) (3) u H = c m u = ρc x y z u (4.) H u = ρc x y z qxyzt (,,, ) qxyzt (,,, ) (4.3) x y z (4.4) (),() x y z x x, yz, f u = k y z (4.5) x x, yz, x x x, yz, y z x x+, yz, f u = k y z (4.6) x x x+, yz, x x+, yz, u u u u k y z + k y z k x z + k x z x x y y x x x x+ y y y y+ u u k x y + k x y z z z z z z+ (4.7)
12 (4.3) (4.4) (4.7) u ρc x y z = qxyzt (,,, ) x y z u u u u + k y z + k x z (4.8) x x x x x + x y y y y y+ y u u k x y z z z z z + z ρc x y z x, y, z 0 0 u u u u κ = κ + + qxyzt (,,, ) + t x y z k (4.9) κ = k cρ
13 u u u u u x xy y x y Axy (, ) + Bxy (, ) + Cxy (, ) + Dxy (, ) + Exy (, ) + Fxyu (, ) = Gxy (, ) B 4AC > 0 B 4AC = 0 B 4AC < 0 Lplce (5.) Φ x Φ y + = 0 (5.) Poisson Φ + + q = 0 x Φ y (5.3) B 4AC = 0 4 = 4 < 0 u u = κ t x (5.4) B 4AC = 0 4 κ 0= 0
14 u u = c t x (5.5) (5.)
15 u u = κ t x (6.) u u u u + u ( x) n+ n n n n i i i+ i i = κ (6.) u u u u + u n+ n n+ n+ n+ i i i+ i i = κ ( x) (6.3) u n+ t=(n+) t i x=i x t x (6.) (6.3)
16 (6.) (6.3) t t n+ n n+ n i- i i+ i- i i+
17 7. x + x + x = b( = ) x + x + x = b( = ) x + x + x = b( = ) (7.) x x + x + x = (7.) x (7.) 3 x + x + x = x + x + x = (7.3) (7.) x x + x = ( ) x + ( ) x = ( ) x + ( ) x = (7.4)
18 (7.4) =, =, = () () 3 () =, =, = () () 3 () (7.5) x + x + x = x + x = () () () x + x = () () () (7.6) () 0 (7.6) () (7.6) x + = (7.7) () () 3 4 x () 3 () (7.6) x3 (7.7) () 3 x + x = (7.8) () () () () 3 () () 3 3 () (7.6) ( ) x = (7.9) () () () () 3 () () () () = = (7.0) () () () () () 3 () () () , () ()
19 x + x + x = x + x = () () () x = () () (7.) x, x, x 3 x () 34 3 = () 33 x = ( x ) () () () (7.) x = ( x x ) (7.) (7.3) () () () = () () () (7.4) ()
20 () () () = () () () () () () 3 () () () 34 3 () (7.5) (7.) x n, n+ n, n+ n n nn nn, + (7.6) n, n+ () () () () 0 3 n, n+ () () () n 3, n+ ( n ) ( n ) nn nn, + (7.7) LU
21 7. 7. (0) Ax= b x. = Ax b A A = A + A A n 3 n = , A = n, n nn n n nn, (7.8) A Α Α (0) (0) Ax b x = A A = nn (7.9) (0) x b b b n x = A b =,,, (7.0) nn (0) T
22 () (0) x x Ax + Ax = b (7.) () (0) () (0) x = A b A Ax (7.) ( k ) ( k = ) x A b A Ax (7.3) ( ) x k i b x = x ( j i) (7.4) ( ) n k i ( k ) i ij ii ii j= SOR
23 Lplce Φ + = 0 x Φ y (8.) Lplce Φi+, j Φ ij, +Φi, j Φij, + Φ ij, +Φij, + = 0 x y ( ) ( ) (8.) Φ ( Φ x + i x, y + j y) ij, 0 0 j + Φ ij+, j Φ i, j Φ ij, Φ i +, j j Φ ij, j i i i i + i +
24 Φ +Φ +Φ +Φ i+, j+ i, j+ i+, j i, j ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x 5 y 5 x y ( Φ +Φ ) + Φ +Φ x y x y + + 0Φ = 0 ij, ( ) i+, j i, j ij, + ij, O( x, y ) O( x 4, y 4 ) x, y Φ ij, ( i = 0,, ie; j = 0,,, je ) Φ ij, ( i = 0,, ie; j = 0,,, je ) Φ Φ n
25 u u = κ (9.) t x 9. u u u u + u n+ n n n n i i i+ i i = κ ( x) (9.) O(, x ) 9. Lsonen Method Lsonen(949) ui ui ui ui + ui = κ x n+ n n+ n+ n+ + ( ) (9.3) O(, x ) Lsonen
26 9.3 Richrdson Method Richrdson Method(90), x + ui ui ui ui + ui = κ x n n n n n + ( ) (9.4) Forsythe & Wsow(960) 9.4 DuFort-Frnkel Method Lepfrog n n+ n Richrdson Method u ( u + u )/ i i i u u u u u + u n+ n n n+ n n i i i+ i i i = κ ( x) (9.5) DuFort & Frnkel(953) Lepfrog O(, x ) 9.5 Crnk-Nicolson Method Crnk & Nicolson(947) u u ( u u + u ) + ( u u + u ) n+ n n n n n+ n+ n+ i i i+ i i i+ i i = κ ( x) (9.6)
27 Crnk-Nicolson Scheme O(, x ) n + n u + i n n ui n u i + n n u i i i i + n + n u + i n u + i n u + i+ n n ui n n u u i i + i i i +
28 u = c t u x (0.) c u u + c = 0 ( c > 0) t x (0.) (0.) (0.) (0.) c (0.) (0.) (0.) 0. Euler n+ n n n ui ui ui+ ui + c = 0 x (0.3) u u u u n+ n n n i i + c i+ i = 0 x (0.4) O(, x) O(, x ) (0.3) (0.4)
29 0. Upwind Difference Method Euler c c u u u u x n+ n n n i i + c i i = 0 (0.5) O(, x) 0.3 Lx Method Euler (0.4) n n n ui ( ui+ + ui )/ Lx Method(954) u ( u + u ) n+ n n i i+ i n n ui+ ui + c = 0 x (0.6) 0.4 Euler u u u u n+ n n + n+ i i i+ i + c = 0 x (0.7) O(, x )
30 0.5 Lep Frog Method Lep Frog Method Lep Frog Method u u u u x n+ n n n i i + c i+ i = 0 (0.8) 0.6 Lw-Wendroff Method Lx-Wendroff Method Tylor n+ n u u ui = ui + t + t + O t t 3 ( ) ( ) (0.9) u = c u, u = c u t x t x (0.0) (0.9) + u u i x x 3 n n u = u c t + c ( t) + O ( ) i (0.) u u x x Lx-Wendroff + ( ) ( ) ( x) n n c n n c n n n i = i i+ i + i+ i + i ( ) u u u u u u u x (0.)
31 0.7 Lx-Wendroff Method Lx-Wendroff n+ / / ( n n ui+ ui+ + ui ) n n ui+ ui + c = 0 / x (0.3) n+ n n+ / n+ / ui ui ui+ ui + c = x / / 0 (0.4) i + / Lep Frog Method Lx-Wendroff Method Lx-Wendroff Method 0.8 McCormck Method McCormck Method(969) McCormck Method Lx-Wendroff Method i + / / i ( ) x n+ n n n ui = ui c ui+ ui (0.5)
32 n+ n n+ n+ n+ ui = ui + ui c ( ui ui ) x (0.6) n u + i n+ n+ u McCormck Method Lx-Wendroff
33 988. C.R G.D Ching-Jen Chen Computtionl Methods in Fluids nd Het Trnsfer, Lecture Note, Deprtment of Mechnicl Engineering, The University of Iow,
34 (3.0) Oh ( ) u u = κ t x u 0 = 0 u 0 =.0 ui( i=,,...,9) 6. (6.3) u u + c = 0 t x McCormck Method Lx-Wendroff Method
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11 Poisson kanenko@mbkniftycom alexeikanenko@docomonejp http://wwwkanenkocom/ , u = f, ( u = u+f u t, u = f t ) 1 D R 2 L 2 (D) := {f(x,y) f(x,y) 2 dxdy < )} D D f,g L 2 (D) (f,g) := f(x,y)g(x,y)dxdy (L
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p P 1 n n n 1 φ(n) φ φ(1) = 1 1 n φ(n), n φ(n) = φ()φ(n) [ ] n 1 n 1 1 n 1 φ(n) φ() φ(n) 1 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 1 4 5 7 8 1 4 5 7 8 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 19 0 1 3 4 5 6 7
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
( ) f a, b n f(b) = f(a) + f (a)(b a) + + f (n 1) (a) (n 1)! (b a)n 1 + R n, R n = b a f (n) (b t)n 1 (t) (n 1)! dt. : R n = b a f (n) (b t
5 1 1.1 ) f, b n fb) = f) + f )b ) + + f n 1) ) n 1)! b )n 1 + R n, R n = f n) b t)n 1 t) n 1)! dt. : R n = f n) b t)n 1 t) n 1)! dt ] b b b t)n 1 + n 1)! = f n 1) b )n 1 ) + R n 1. n 1)! R n = [f n 1)
N N 1,, N 2 N N N N N 1,, N 2 N N N N N 1,, N 2 N N N 8 1 6 3 5 7 4 9 2 1 12 13 8 15 6 3 10 4 9 16 5 14 7 2 11 7 11 23 5 19 3 20 9 12 21 14 22 1 18 10 16 8 15 24 2 25 4 17 6 13 8 1 6 3 5 7 4 9 2 1 12 13
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
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) du (L) = f (9.3) dx (9.) P
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)
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A B,BW C Rc No. h Rc C 2 3 4 M2 M3 M4 M5 M6 25 3.2 6.8 3.5 6.4 2.8 2.7 2.4 9.1 15.5 21.6 28.0 34.4 35 41 40 50 60 80 100 120 140 160 180 200 240 4.8 10.2 5.2 10.1 4.3 4.1 3.8 14.23 24.36 34.1 44.2 54.3
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