Método de Crank - Nicholson

 

Mêtlgl gi Drnc` — Cjdalfslc

 

Giguddjôc gi fn idundjôc gi fn lcgn (Ulr gjbiricdjns cjtns)  

(>)

(>) si ffnmn ucn idundjôc gjbiricdjnf pnrdjnf pnrnkôfjdn. pnrnkôfjdn. Ycl gi fls ihimpfls sjmpfis gi ucn IGU pnrnkôfjdn is fn idundjôc gi dlcguddjôc gi dnflr pnrn ucn knrrn gi mitnf si muistrn ic fn ourn J n dlccundjôc0

(;)

Glcgi b = timpirnturn dlml ucn bucdjôc gi fn ukjdndjôc ‖x ‖y impl ‖t ‖ic if qui fn gjbusjvjgng têrmjdn ΰ si gn plr 8 ξ = dlcgudvjgng têrmjdn gif mntirjnf gi fn knrrn ώ=Gicsjgng gif mntirjnf gi fn knrrn. D= dnflr ispidídl gif mntirjnf gi fn knrrn

 

If Mêtlgl gi fns Gjbiricdjns Bjcjtns, dlcsjsti ic jctrlgudjr ucn mnffn slkri ucn rriojôc iojôc y nprl nprlxjmnr xjmnr fns girjvngns gif prlkfimn pfnctingl plr migjl gi têdcjdns gi nprlxjmndjôc gi fns girjvngns dlml fl is, plr ihimpfl, fn gisdlmplsjdjôc ic sirji gi Wnyflr Wnyflr gi fn bucdjôc.

 

Fn blrmn gi gjbiricdjns cjtns gi fn idundjôc gi dnflr (;) nalrn si gn dlc fn sioucgn girjvngn ispndjnf ivnfungn n pnrr gi ucn dlmkjcndjôc gi fls girjvngls ic pnsls gi impl c y c + >

(4)

Gi fn bôrmufn nctirjlr, vnmls n ticir uc isquimn ixpfídjtl duncgl m = > y uc isquimn jmpfídjtl duncgl m = 5, dlml si muistrn n dlccundjôc ic fns idundjlcis (7) y (9) ( 9) rispidvnmicti8 rispidvnmicti8 (7)

(9)

 

Gi aidal m puigi sir dunfqujir vnflr ictri 5 y >, sjc imknrol ucn blrmn lpdjôc dlmøc is m= 5,9. (4) si vuifvi0 (2)

Fn idundjôc nctirjlr (2) puigi sir dlfldngl gi mncirn qui fns timpirnturns ic if jctirvnfl gi impl prisicti (c + >) istîc ic if fngl jzqujirgl. jzqujir gl. Migjncti fn npfjdndjôc gi istns idundjlcis pnrn tlgls fls clgls, lkticgrimls uc sjstimn dlc mntrjz gi dlidjictis trjgjnolcnf.

(?)

 

Fn Ixprisjôc si dlcldi dlml if cømirl gjbusjôc y si gicltn plr r= n •   dlccundjôc, (?) si dlcvjirti

(1)

Ulr fl tnctl8

(:)

(:) si ffnmn n fn nprlxjmndjôc Drnc` Cjdlfslc

 

IHI[DJDJL.- Ycn vnrjffn gi ndirl si slmiti n ucn timpirnturn gi >55µD ic if ixtriml jzqujirgl y ;9µD ic if ixtriml giridal. Tj fn knrrn ici ucn flcojtug gi 5,59 m, ufjdi if mêtlgl gi Drnc`-Cjdlfslc pnrn icdlctrnr fn gjstrjkudjôc gi fn timpirnturn ic fn knrrn gi t = 5 n t = : sioucgls. Yfjdi ηx= 5,5> m y ηt = 4s. Gngl8

Fn timpirnturn jcjdjnf gi fn knrrn is ;5µD Tlfudjôc

Ictlcdis

 

Bjourn >.- Gjnornmn isquimîdl qui muistrn fn riprisictndjôc gi clgls ic if mlgifl

(:) Fn timpirnturn jcjdjnf gi fn knrrn is ;5 µD, is gidjr, tlgns fns timpirnturns gi fls clgls gictrl gi fn vnrjffn istîc ic ;5 D (t=5), n ixdipdjôc gi fls clgls brlctirn gngns plr (:). Istl plgrín sir riprisictngl dlml Unrn j8 >,;,4,7.

(>5)

Fn timpirnturn jcjdjnf ic fls clgls gictrl gi fn vnrjffn (duncgl t =5 sioucgls) istî gngn plr (:) (>5) (:)

 

sioucgls,, n pnrr gi (>), Fn timpirnturn ic fls clgls gictrl gi fn vnrjffn duncgl t= 4 sioucgls ticimls fn dlcgjdjôc gi dlctlrcl gi fn blrmn 8 Gi fn dlcgjdjôc (>) Unrn tlgls fls clgls jctirjlris, istnkfidjicgl c = 5 y j = >,;,4,7 ic (:) gn fls sjoujictis idundjlcis

 

Fn mntrjz gi dlidjictis ic if nctirjlr dlchuctl gi idundjlcis is trjgjnolcnf. Fn rislfudjôc gi istn mntrjz is fn sjoujicti8

fn timpirnturn ic tlgls fls clgls ic if impl,  impl, t = 4 sioucgls.

 

Fn timpirnturn ic fls clgls gictrl gi fn vnrjffn duncgl t = 2 sioucgls8 Gi fn dlcgjdjô dlcgjdjôc c (>) Unrn tlgls fls clgls jctirjlris, istnkfidjicgl c = > y j = >,;,4,7 ic (:) gn fls sjoujictis idundjlcis

 

Gi jounf mncirn qui ic if nctirjlr dnsl.

fn timpirnturn ic tlgls fls clgls ic if impl, t = 2 sioucgls.

Fn timpirnturn ic fls clgls gictrl gi fn vnrjffn duncgl t = : sioucgls8 Gi fn dlcgjdjôc (>) Unrn tlgls fls clgls jctirjlris, istnkfidjicgl c = ; y j = >,;,4,7 ic (:) gn fls sjoujictis idundjlcis

 

Gi jounf blrmn qui ic if nctirjlr dnsl.

 

fn timpirnturn ic tlgls fls clgls ic if impl, t=: sioucgls.

tnkfn j8 j 8 Wimpirntur Wimpirnturn n lkticjgl ic jctirjlr clgls c = ; y j =>,;,4,7 ufjzncgl if mêtlgl drnc` cjdlfslc Wimpirnturn ic fls clgls

Mêtlgl Drnc` Cjdlfslc

b4 >

2;.257

b4 ;

4?.2>4

b4 4

;2.92;

b4 7

;7.57;