## Question

###### Finite [Ji( (crence Apprximariont Tno Spice Dimtnsian(orm of (4.9.1). Thus we Yrilc with 0< 8, < | and 0<8*1.J M (4.9,2)(. 2v,." ",+l,"-14'(4,t" 26,.4.1 WL)(u;.1 26,." "1,-1)(7" 20,.1' "-1).bclore,selectwc shall obtain approximation (termed the classic explicit)explicit finile difference(4.9.)0(", ",-" 06.3014,,4)[classic explicit]Note that (4.9.3) uses the five points laheled 0, 1, 2, % and in Figure I0 t0 calculate Obvio

Finite [Ji( (crence Apprximariont Tno Spice Dimtnsian (orm of (4.9.1). Thus we Yrilc with 0< 8, < | and 0<8*1. J M (4.9,2) (. 2v,." ",+l,"-14' (4,t" 26,.4.1 WL) (u;.1 26,." "1,-1) (7" 20,.1' "-1). bclore, select wc shall obtain approximation (termed the classic explicit) explicit finile difference (4.9.) 0(", ",-" 06.301 4,,4) [classic explicit] Note that (4.9.3) uses the five points laheled 0, 1, 2, % and in Figure I0 t0 calculate Obviously. the method of cakculalion involves point-by - point evaluation plane using Ihe points on the planc , Given the initial condilions for the plane by plane evaluation follows As special case the choice yields (4.9.4) ",44,64 46 JMi ",..-) we thercby eliminale the MeTi Stability Ihe classic explicit approximation (4.9.3) can Acetaaincd using von Neumann" mcthod: The extension of (4.5.5) for the one-dimensiona cusc nov bccomes Elx,Y. Y)ze"e" "e When subsliluled into (4.9.3) and common lerms canceled, there results (4.9.51 amplification (actor = &** = | ~ 4p(sin? Bk +sin? 44 Since |e/ <1 required for stability, follows that ~Isl-4p (sin? B* +sin" @#)si Pumbolic Panitl Dillarentinl Fquation But 8, and B, are arbilrary and follows thal (4.9.6) P < 2(sin? Bzk +sin'&k We scc thal (4.9,4) is taken thc uppcr limit of the stability bound (or (4.9.3). Finally: had not uscd k =kz but relained k + kz- the stability bound would b (4,9.7) h<- V