Numerical Method: Finite difference scheme
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- Terence Harper
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1 Numrcal Mthod: Ft dffrc schm Taylor s srs f(x 3 f(x f '(x f ''(x f '''(x...(1! 3! f(x 3 f(x f '(x f ''(x f '''(x...(! 3! whr > 0 from (1, f(x f(x f '(x R Droppg R, f(x f(x f '(x Forward dffrcg O ( x from (, f(x - f(x f '(x ackward dffrcg O ( x Substractg ( from (1 : f(x - f(x - f '(x... f '(x f(x - f(x - Addg(1 ad ( Ctral dffrcg O ( x f(x - f (x f(x - f ''(x
2 Cosdr th smplst quato.. advcto quato F F c 0, c > 0...(3 t x x π Lt Lt F(x,0 A whr L Applyg sparato of varabls tchqu to (3 F(x, t G(tH(x G H H(x cg(t 0 t x 1 G 1 H c k (say G t H x G - k G(t t dg k dt l G kt costat G G Lt us assum G A at t o l kt A kt 1 or G A1 1 H k From th othr quato H(x; o gts H x c A kx c
3 F(x, t A A -kt Ital Codto : F(x,0 If A1A A, Soluto s; 1 th F(x, t kx u A x k c x A ( ct Ths s th Aalytcal Soluto
4 Lar computatoal Istablty Cosdr th advcto quato F F c t x 0 c > 0 (1 m 0, ± 1, ±,... 0,1,... t ; x m Aalytcal soluto s F(x, t A ( xct whr π L Numrcal quatos corrspodg to (1 th Ctral dffrc schm F m, 1 Fm, 1 Fm c F 1, F m-1, 0 m F m, A Hr may b complx. ( (3 (4
5 Substtutg (4 to (3 gvs, { ( 1 Δ t ( 1 Δ t m Δ x Δ t ( m Δ x ( m Δ } { 1 1 x } ( ( Multplyg through by t Δ c Δ t Δ t t 1 ( S σ 1 0 whr, σ Solvg th quadratc (5 o gts, (5 ( S (6 σ ± 1σ (7 Two cass may ars
6 Stabl Cas: : σ 1 o xpotal growth. I ths cas th radcal s ral, ad th two valus of t may b wrtt polar form as, Δ ( π ad (8 whr arc Sσ ( π Cos S Cos S Hc from (4 [ ( ] π m F M E (9 m, whr M ad E ar arbtrary costats to b dtrmd by tal codtos. At tal tm 0. Hc from (9 [ M E] m Fm,0
7 m From (9 F [ M E] m,0 π Hc MEA, Also 1 So from (9 F m, or, F D M ( m Δ x / π ( m Δ x / E ( m / ( m / ( A E E( 1 (10 W dsgat ths ft dffrc soluto by th subscrpt D. Comparg ths soluto wth th aalytcal soluto rvals som mportat dffrcs. Frst, thr ar two wavs th ft dffrc soluto F D, whr as thr s oly a sgl wav th tru soluto F. Th two wavs, or mods, occur bcaus th ft dffrc quato s of d ordr, whch lads to costats dotd by M ad E (9.
8 Thus M s lmatd by th tal codto. Th d costat E wll b dtrmd th frst tm stp th followg way. Th smplst procdur s to tak a forward stp tm whl rtag a F ctrd dffrc for. x Fm,1 m,0 m m 1,0 F F c 1,0 Δ Substtutg th tal valus F x 0 F m,0 A m Δ x m x F m,1 A A c Δ ( m 1 ( m A x Δ 1 0
9 or, F m m -m m,1 A 1 ( A m 1 A(1 σ s m m A (1 s (11 Ths s th valu assgd to th soluto (10 for 1, Hc, A( 1- s or, A( or, E( or, E or, E 1 s cos - m Δ x m x- m - (A (A E E A( 1 - s A( 1 cos A( 1- cos cos Thus th scod costat of tgrato has b dtrmd by spcfyg th soluto at t t addto to t0. E - E
10 Substtutg ths valu of E (10 gvs th complt soluto to th dffrtal quato. m m A(1 cos ( 1 (1 cos Fm, (-1 A cos cos hr s c s ( s ( As 0, 1. s c I addto for small, s -1 ( c c (1
11 Thus th xpotal trm bcoms: m ± ( m ± lttg 0, 0 1 cos (1 cos 1, cos cos 0 Thus as ad td to zro, kpg 1 th frst trm (1 approachs th tru soluto whl scod trm vashs. Ths s calld th Courat Frdrchs Lwy codto to avod Computato stablty λ x π Hr wll dcras wth dcrasg L Thus wh th wavlgth of th dsturbac s larg compard to th grd lgth, th ampltud of th physcal mod wll b arly that of th aalytcal wav ad th ampltud of th computatoal mod wll b small.
12 Wh λ 0.75, For ( 0. 1L th ampltud of th physcal mod s about 0.06A o th 1.06A othrhad ad that wh ( of computatoal mod s about ( rlatvly l arg π Δ x 0.5L, Hr th ampltud of th physcal mod s zro ad th spurous mod has ampltud A, whch s dsastrous. It was otd arlr that for λ 1,. cδ t th phas spd of th physcal mod s dtcal to that of th aalytcal wav soluto.
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