Partial Differential Equations part 1

... representation (see Figure 19 .0.2), u j +1, l − 2u j,l + u j 1, l ∆ 2 + u j,l +1 − 2u j,l + u j,l 1 ∆ 2 = ρ j,l (19 .0.5) or equivalently u j +1, l + u j 1, l + u j,l +1 + u j,l 1 − 4u j,l =∆ 2 ρ j,l (19 .0.6) To write ... • • 1    • −4 1     1 −4    −4 1 1 −4 •   1 • •   • • 1    • −4 1     1 −4    •   •    •     •   ...

Ngày tải lên: 28/10/2013, 22:15

... (Figure 19 .1. 2): u n j → 1 2  u n j +1 + u n j 1  (19 .1. 14) This turns (19 .1. 11) into u n +1 j = 1 2  u n j +1 + u n j 1  − v∆t 2∆x  u n j +1 − u n j 1  (19 .1. 15) 838 Chapter 19 . Partial Differential ... equation (19 .1. 20) is r n +1 j +1/ 2 − r n j +1/ 2 ∆t = s n +1/ 2 j +1 − s n +1/ 2 j ∆x s n +1/ 2 j − s n 1/ 2 j ∆t = v r n j +1/ 2 − r n j 1/ 2 ∆x (19...

Ngày tải lên: 07/11/2013, 19:15

... drivers. Section 16 .6 of this chapter treats the subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19 ). ... conditions 707 708 Chapter 16 . Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-5 21- 4 310 8-5) Copyright...

Ngày tải lên: 15/12/2013, 04:15

... slight generalization of (19 .2.8) leads to i  ψ n +1 j − ψ n j ∆t  = −  ψ n +1 j +1 − 2ψ n +1 j + ψ n +1 j 1 (∆x) 2  + V j ψ n +1 j (19 .2.27) for which ξ = 1 1+i  4∆t (∆x) 2 sin 2  k∆x 2  + V j ∆t  (19 .2.28) This ... schemes: u n +1 j − u n j ∆t = D 2  (u n +1 j +1 − 2u n +1 j + u n +1 j 1 )+(u n j +1 − 2u n j + u n j 1 ) (∆x) 2  (19 .2 .13 ) Here both the lef...

Ngày tải lên: 15/12/2013, 04:15

... is u n +1 j,l = 1 4 (u n j +1, l + u n j 1, l + u n j,l +1 + u n j,l 1 ) − ∆t 2∆ (F n j +1, l − F n j 1, l + F n j,l +1 − F n j,l 1 ) (19 .3.3) Note that as an abbreviated notation F j +1 and F j 1 refer ... is second-order accurate and unitary: e −iHt  1 − 1 2 iH∆t 1+ 1 2 iH∆t (19 .2.35) In other words,  1+ 1 2 iH∆t  ψ n +1 j =  1 − 1 2 iH∆t  ψ n j (19 .2.36)...

Ngày tải lên: 15/12/2013, 04:15

... have u H jl = 2 L L 1  n =1 A n sinh πnj J sin πnl L (19 .4 .13 ) 860 Chapter 19 . Partial Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-5 21- 4 310 8-5) Copyright ... f l (19 .4 .17 ) The model equation (19 .0.3) becomes ∇ 2 u  = −∇ 2 u B + ρ (19 .4 .18 ) or, in finite-difference form, u  j +1, l + u  j 1, l + u  j,l +1...

Ngày tải lên: 15/12/2013, 04:15

... In matrix notation, equations (19 .5.35) are (L x + r1) · u n +1/ 2 =(r1−L y )·u n −∆ 2 ρ (19 .5.36) (L y + r1) · u n +1 =(r1−L x )·u n +1/ 2 − ∆ 2 ρ (19 .5.37) where r ≡ 2∆ 2 ∆t (19 .5.38) The matrices ... as x (r) = x (r 1) − (L + D) 1 · [(L + D + U) · x (r 1) − b] (19 .5 .16 ) The term in square brackets is just the residual vector ξ (r 1) ,so x (r) =x (r 1) − (L + D) 1 · ξ (r...

Ngày tải lên: 15/12/2013, 04:15

... (19 .6 .15 ) and H =2h. You will find that (Rv h ) (x,y) = 1 4 v h (x, y)+ 1 8 v h (x+h, y)+ 1 16 v h (x + h, y + h)+··· (19 .6 .16 ) so that the symbol of R is    1 16 1 8 1 16 1 8 1 4 1 8 1 16 1 8 1 16    (19 .6 .17 ) Note ... ngrid]=dmatrix (1, nn ,1, nn); rstrct(irho[ngrid],irho[ngrid +1] ,nn); } nn=3; iu [1] =dmatrix (1, nn ,1, nn); irhs [1] =dmatrix (1, nn...

Ngày tải lên: 24/12/2013, 12:16

... of Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-5 21- 4 310 8-5) Copyright (C) 19 88 -19 92 by Cambridge University Press.Programs Copyright (C) 19 88 -19 92 ... interval x1,x2 . Query for another plot until the user signals satisfaction. { int jz,j,i; float ysml,ybig,x2,x1,x,dyj,dx,y[ISCR +1] ; char scr[ISCR +1] [JSCR +1] ; for (;;) { printf...

Ngày tải lên: 07/11/2013, 19:15

... Seinfeld, J. 19 71, Numerical Solution of Ordinary Differential Equations (New York: Academic Press). 16 .1 Runge-Kutta Method The formula for the Euler method is y n +1 = y n + hf(x n ,y n ) (16 .1. 1) whichadvances ... across the whole interval. Figure 16 .1. 2 illustrates the idea. In equations, k 1 = hf(x n ,y n ) k 2 =hf  x n + 1 2 h, y n + 1 2 k 1  y n +1 = y n +...

Ngày tải lên: 15/12/2013, 04:15