Explicit scheme. Fully implicit scheme Notes. Fully implicit scheme Notes. Fully implicit scheme Notes. Notes

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1 Explicit cheme So far coidered a fully explicit cheme to umerically olve the diffuio equatio: T + = ( )T + (T+ + T ) () with = κ ( x) Oly table for < / Thi cheme i ometime referred to a FTCS (forward time cetered pace) It i fully explicit, ice T + ca be computed from kow quatitie at time Advatage: Eay to olve umerically, becaue it explicit Diadvatage: Uually itereted i feature of ize λ x Let t λ be the time to diffue a ditace λ, which i λ /κ I order to atify tability criterio, eed t λ / (λ /κ)/( x /κ) λ / x time tep before thig happe o the cale of iteret Computatioally expeive/prohibitive Fully implicit cheme There are other way to dicretize the diffuio equatio Coider the followig differece cheme: T + T = κ T T + T + ( x) () Thi i very imilar to FTCS, except that the patial derivative o the RHS i evaluated at time tep ( + ), ot Thi cheme i called fully implicit or backward time I cotrat to the fully explicit cheme, T = caot be olved purely i term of fuctio value at time tep Fully implicit cheme Why i it called backward time? Rearragig yield T = T + where = κ/ x + ( + )T + T + + (3) Therefore, oe ca obtai (explicitly) T i term of value of the ext time tep ( + ) Thi i ot what we wat, however Fully implicit cheme I order to obtai the fuctio value at time tep ( + ) eed to olve a et of imultaeou liear equatio (eq(3)), which ca be cat i matrix form: ( + ) T + T ( + ) T + T T + = T ( + ) T + T J J

2 Fully implicit cheme Do matrix iverio to obtai the fuctio value at time tep ( + ) T + ( + ) T + ( + ) T + = T + J ( + ) T T T Sice matrix i tridiagoal, efficiet algorithm exit to ivert matrix Alo, if κ i a cotat, i cotat ad iverio oly ha to be doe oce T J Fully implicit cheme I it table? Let do vo Neuma tability aalyi: Subtitute T = ξ e ik x ito the differece cheme: T = T ( + )T T + + ξ e ik x = ξ + e ik( ) x + ( + )ξ + ξ + e ik(+) x = ξe ik x + ( + )ξ ξe ik x ξ = + (e ik x + e ik x ) ξ = + ( co(k x)) ξ = + 4(i(k x/)) Sice i, ξ for all k, the fully implicit cheme i ucoditioally table Improvig upo the implicit cheme Eve though the fully implicit cheme i ucoditioally table, the accuracy ha ot improved While it i ecod order accurate i the patial part, it i oly firt order accurate i the temporal part We ca improve the fully implicit cheme by averagig the explicit ad implicit differece cheme: where T + T = κ [(δ T ) + + (δ T ) ] (4) (δ T ) = T + T + T ( x) Both the LHS ad RHS are ow cetered aroud ( + /), which make the cheme ecod order accurate i time Thi cheme i kow a the Crak Nicholo cheme Stability of Crak-Nicholo How table i it? Subtitutig T = ξ e ik x ito the differece cheme yield a amplificatio factor which i for all k ξ = ( i( k x )) + ( i( k x )) (5) So the Crak-Nicholo cheme ha ucoditioal tability ut like the fully implicit cheme I additio it i ecod order accurate i both time ad pace I therefore the recommeded method for thee type of PDE

3 Computatioal implemetatio Sice the cheme i ot fully explicit, we eed to olve a et of coupled liear equatio T + T T + T = κ [(δ T ) + + (δ T ) ] [ = κ (T T + T + ) + (T + T + T ( x) ) ] Regroupig the term yield T ( + )T T + = T + ( )T + T Computatioal implemetatio Agai, we ca cat the coupled equatio ito a matrix: ( + ) ( + ) ( + ) J ( ) T ( ) = T T ( ) T J T + T + T + T + Multiplyig by the ivere of the firt matrix allow u to compute the T + Summary FTCS (fully explicit):firt order accurate i time, ecod order accurate i pacecoditioally table: κ/ x /Eve though eay to implemet, the tability criterio impoe mall time tep, which i computatioally exteive Fully implicit:ucoditioally table, but accuracy i the ame a for FTCS Crak Nicholo:Combie the fully implicit ad explicit chemethe patial ad time derivative are both cetered aroud + /Therefore, the method i ecod order accurate i time (ad pace)ucoditioally table Crak Nicholo i the recommeded method for olvig diffuive type equatio due to accuracy ad tability Higher dimeio So far coidered oly oe patial dimeio for implicity Exteio to higher dimeio i traightforwardcoider the diffuio equatio i two dimeio: ( T ) t = κ T x + T y Approximatig T (x, y, t) T ( x, l y, ) ad implemetig the Crak-Nicholo cheme give u T +,l T,l = κ [(δ xt ),l + (δ xt ) +,l + (δy T ),l + (δ y T ) +,l ] where ad imilarly for (δ y T ),l (δ xt ),l = T +,l T,l + T,l ( x)

4 Higher dimeio Note: Eve though the coupled liear equatio ca be cat ito a matrix that i pare, it i ot tridiagoal aymore a for the oe dimeioal cae Therefore, computatio time ca icreae igificatly Note: Method exit to circumvet thi problem (ee Numerical recipe book) Numerically olvig the time-depedet Schrodiger equatio The time-depedet Schrödiger equatio ha imilar tructure a the diffuio equatiohowever, it i ot diffuive, but diperive Solutio ted to break up ito ocillatory wave packet I oe patial dimeio the equatio read (with = ad m = /) i ψ = Hψ (6) t where H = x + V (x) If oe i give the (ormalized) iitial wave packet ψ(x, t = ), we may ue the fiite differece cheme we developed for the diffuio equatio(eg BC i that ψ a x ± )We the umerically itegrate the Schrödiger equatio i order to fid the wave fuctio ψ(x, t) at later time Implicit cheme for the Schrödiger equatio Aalogou to the heat equatio we ca apply the implicit differece cheme i ψ+ ψ = ψ+ + ψ+ + ψ + ( x) + V ψ + (7) The vo Neuma tability aalyi yield Therefore, ξ = [ + i 4 i ( ) (8) k x ( x) + V ] ξ = [ + 4 i ( k x ( x) ) ] (9) + V Implicit cheme for the Schrödiger equatio Depite the ucoditioal tability of the implicit cheme, it i ot appropriate for olvig the Schrödiger equatio The reao i that the wave fuctio ψ eed to remai ormalized at all time durig the time evolutio: + ψ dx = () If the iitial coditio/wave fuctio ψ(x, t = ) i ormalized, the the Schrödiger equatio eure thi ormalizatio coditio durig the time evolutio of the wave packet

5 Uitary requiremet Formally, thi ca be how Let itegrate the Schrödiger equatio, i ψ = Hψ () t where the Hamiltoia operator i H = x + V (x) The formal olutio i imply ψ(x, t) = e iht ψ(x, ) ψ(t) = e iĥt ψ() bra-ket otatio ψ(t) = ψ() e iĥt the cougate Note, that Ĥ i elf-adoit Ĥ + = Ĥ ad uitary HH + = H + H = I Uitary requiremet Normalizatio i guarateed, ice the time evolutio operator e iĥt i uitary + ψ dx = ψ(t) ψ(t) = ψ() e iĥt e iĥt ψ() = ψ() ψ() = The implicit cheme approximate the time evolutio ψ(x, t) = e iht ψ(x, ) a ψ(x, t) = ψ + ψ(x, ) eiht + ih ψ But the approximatio of the time evolutio operator ( + ih) i ot uitary Uitary requiremet Let how briefly that thi approximatio i ideed the implicit cheme: where Therefore, we have i ψ+ ψ ( + ih)ψ + = ψ ψ + ψ = ihψ + Hφ = ψ + ψ + ψ ( x) which i equivalet to eq (7) + V ψ = ψ+ + ψ+ + ψ + ( x) + V ψ + Caley form The FTCS cheme uffer from the ame problem Here, the time evolutio i approximated a ψ(x, t) = e iht ψ(x, ) ψ + ( ih)ψ Agai, ( ih) i ot uitary We ca remedy thi problem by uig Caley form for the fiite differece approximatio of e iht : ψ(x, t) = e iht ψ(x, ) ψ(x, t) = e iht/ ψ(x, ) eiht/ ψ + ( ih/) ( + ih/) ψ

6 Caley form The approximatio of the time evolutio operator e iht ( ih/) ( + ih/) i uitary The differece cheme i the ( + ih/)ψ + = ( ih/)ψ () which i ucoditioally table, uitary ad ad ecod order accurate i pace ad time I fact it i the Crak-Nicholo cheme Rearragig yield i ψ+ ψ = (Hψ+ + Hψ ) (3) Summary The Schrödiger equatio i a example where tability of the umerical olutio aloe i ot ufficiet to obtai good reult We alo wat the umerical cheme to obey the uitarity requiremet, uch that the total probability remai It happe that the Crak-Nicholo cheme doe ut that It iterpolate the Hamiltoia betwee time tep (FTCS) ad ( + ) (implicit) Example - applet: