U t + u U x µ 2 U = 0. (101)

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2 Chapter 3 Fte Dfferece Methods I the prevous chapter we developed fte dfferece appromatos for partal dervatves. I ths chapter we wll use these fte dfferece appromatos to solve partal dfferetal equatos (PDEs) arsg from coservato law preseted Chapter. 48 Self-Assessmet Before readg ths chapter, you may wsh to revew... Coservato Laws Fte Dfferece Appromatos 2 After readg ths chapter you should be able to... mplemet a fte dfferece method to solve a PDE compute the order of accuracy of a fte dfferece method develop upwd schemes for hyperbolc equatos Relevat self-assessmet eercses: Fte Dfferece Methods Cosder the oe-dmesoal covecto-dffuso equato, U t + u U µ 2 U = 0. (0) 2 Appromatg the spatal dervatve usg the cetral dfferece operators gves the followg appromato at ode, du dt + u δ 2 U µδ 2 U = 0 (02) Ths s a ordary dfferetal equato for U whch s coupled to the odal values at U ±. Assemblg all of the odal states to a sgle vector gves a system of coupled ODEs of the form: U =(U 0, U,..., U, U, U +,..., U N, U N ) T, 67

3 68 du dt = AU+ b (03) where b wll cota boudary codto related data (boudary codtos are dscussed Secto 49.). The matr A has the form: A 0,0 A 0,... A 0,N A,0 A,... A,N A = A N,0 A N,... A N,N Note that row of ths matr cotas the coeffcets of the odal values for the ODE goverg ode. Ecept for rows requrg boudary codto data, the values of A, j are related to the coeffcets α j of the fte dfferece appromato. Specfcally, for our cetral dfferece appromato A, = u 2 + µ 2, A, = 2 µ 2, A,+ = u 2 + µ 2, ad all other etres row are zero. I geeral, the umber of o-zero etres row wll correspod to the sze of the stecl of the fte dfferece appromatos used. We refer to Equato 03 as beg sem-dscrete, sce we have dscretzed the PDE space but ot tme. To make ths a fully dscrete appromato, we could apply ay of the ODE tegrato methods that we dscussed prevously. For eample, the smple forward Euler tegrato method would gve, U + U t = AU + b. (04) Usg cetral dfferece operators for the spatal dervatves ad forward Euler tegrato gves the method wdely kow as a Forward Tme-Cetral Space (FTCS) appromato. Sce ths s a eplct method A does ot eed to be formed eplctly. Istead we may smply update the soluto at ode as: U + Eample. Fte Dfferece Method appled to -D Covecto I ths eample, we solve the -D covecto equato, = U t (u δ 2 U µδ 2 U ) (05) U t + u U = 0, usg a cetral dfferece spatal appromato wth a forward Euler tme tegrato, U + t U + u δ 2 U = 0. Note: ths appromato s the Forward Tme-Cetral Space method from Equato wth the dffuso terms removed. We wll solve a problem that s early the same as that Eample 3. Specfcally, we use a costat velocty, u= ad set the tal codto to be U 0 ()=0.75e ( ) 2. We cosder the doma Ω =[0,] wth perodc boudary codtos ad we wll make use of the cetral dfferece appromato developed Eercse. The matlab scrpt whch mplemets ths algorthm s:

4 69 % Ths Matlab scrpt solves the oe-dmesoal covecto 2 % equato usg a fte dfferece algorthm. The 3 % dscretzato uses cetral dffereces space ad forward 4 % Euler tme. 5 6 clear all; 7 close all; 8 9 % Number of pots 0 N = 50; = lspace(0,,n+); 2 d = /N; 3 4 % velocty 5 u = ; 6 7 % Set fal tme 8 tfal = 0.0; 9 20 % Set tmestep 2 dt = 0.00; % Set tal codto 24 Uo = 0.75*ep(-((-0.5)/0.).ˆ2)'; 25 t = 0; U = Uo; % Loop utl t > tfal 30 whle (t < tfal), 3 % Forward Euler step 32 U(2:ed) = U(2:ed) - dt*u*cetraldff(u(2:ed)); 33 U() = U(ed); % eforce perodcty % Icremet tme 36 t = t + dt; % Plot curret soluto 39 clf 40 plot(,uo,'b*'); 4 hold o; 42 plot(,u,'*','color',[ ]); 43 label('','fotsze',6); ylabel('u','fotsze',6); 44 ttle(sprtf('t = %f\',t)); 45 as([0,, -0.5,.5]); 46 grd o; 47 drawow; 48 ed Fgure 20 plots the fte dfferece soluto at tme t= 0.25, t= 0.5, ad t=.0. The eact soluto for ths problem has U(,t)= U o () for ay teger tme (t =,2,...). Whe the umercal method s ru, the Gaussa dsturbace covected across the doma, however small oscllatos are observed at t = 0.5 whch beg to pollute the umercal soluto. Evetually, these oscllatos grow utl the etre soluto s cotamated. I Chapter 4 we wll show that the FTCS algorthm s ustable for ay t for pure covecto. Thus, what we are observg s a stablty that ca be predcted through some aalyss. Eercse. Dowload the matlab code from Eample ad modfy the code to use the backward dfferece formula δ. Ths method kow, as the Forward Tme-Backward Space (FTBS) method. Usg the same u=, t = 000 ad = 50 does the FTBS method ehbt the same stablty as the FTCS method?

5 70 t = t = t = U o U(t) U o U(t) U o U(t) U 0.5 U 0.5 U (a) t = (b) t = (c) t =.00 Fg. 20 Forward Tme-Cetral Space method for -D covecto. (a) Yes (b) No (c) Sometmes (d) I do t kow 49. Boudary Codtos I ths secto, we dscuss the mplemetato of fte dfferece methods at boudares. Ths dscusso s ot meat to be comprehesve, as the ssues are may ad ofte subtle. I partcular, we oly focus o Drchlet boudary codtos. A Drchlet boudary codto s oe whch the state s specfed at the boudary. For eample, a heat trasfer problem the temperature may be kow at the doma boudares. Drchlet boudary codtos ca be mplemeted a relatvely straghtforward maer. For eample, suppose that we are solvg a oe-dmesoal covecto-dffuso problem ad we wat the value of U at =0, to be U let, U 0 = U let. To mplemet ths, we f U 0 = U let ad apply the fte dfferece dscretzato oly over the teror of the computatoal doma accoutg for the kow value of U 0 at ay place where the teror dscretzato depeds o t. For eample, at the frst teror ode (.e. = ), the cetral dfferece dscretzato of for the -D covecto-dffuso equato gves, du dt + u U 2 U 0 2 Accoutg for the kow value of U 0, ths becomes, du dt + u U 2 U let 2 = µ U 2 2U +U 0 2. = µ U 2 2U +U let 2. (06) I terms of the vector otato, whe a Drchlet boudary codto s appled we usually remove that state from the vector U. So, the stuato where U 0 s kow, the state vector s defed as, U =(U,U 2,..., U,U,U +,..., U N,U N ) T,

6 The b vector the wll cota the cotrbutos from the kow boudary values. For eample, by re-arragg Equato (06), the frst row of b cotas, b = u 2 U let 2 + µ U let 2. Sce U let does ot eter ay of the other ode s stecls, the remag rows of b wll be zero (uless they are altered by the other boudary). 7 Eercse 2. Dowload the matlab code from Eample ad modfy the code to use a Drchlet boudary codto o the flow ad the backwards dfferece formula δ o the outflow. Usg the same u=, t = 000 ad = 50 s the FTCS method wth Drchlet boudary codto stable? (a) Yes (b) No (c) Sometmes (d) I do t kow 49.2 Trucato Error for a PDE I the dscusso of ODE tegrato, we used the deas of cosstecy ad stablty to prove covergece through the Dahlqust Equvalece Theorem. Smlar cocepts also est for PDE dscretzatos, however, we caot cover these here. We wll brefly look at the trucato error for a partal dfferetal equato. We have already dscussed the local trucato error for fte dfferece appromato of dervatves Secto 47.. Smlar to the ODE case, the trucato error s defed as the remader after a eact soluto to the goverg equato s substtuted to the fte dfferece appromato. For eample, suppose we are usg the FTCS algorthm Equato to appromate the oe-dmesoal covecto-dffuso equato. The the local trucato error for the PDE appromato s defed as, τ U + t U + u δ 2 U µδ 2 U, (07) where U(,t) s a eact soluto to Equato (94). Note that the trucato error defed here for PDE s s ot qute a drect aalogy wth the stadard defto of local trucato error used ODE tegrato, specfcally Equato (6). I partcular, the ODE case, the trucato error s defed as the dfferece betwee the umercal soluto ad the eact soluto after oe step of the method (startg from eact values). However, the PDE case, we have defed the trucato error as the remader after a eact soluto s substtuted to the umercal method whe the umercal method s wrtte a form that appromates the goverg PDE. Ecept for ths dfferece the defto, the calculato of the local trucato follows the same procedure as the ODE case whch Taylor seres substtutos are used to epad the error powers of t. Cotug o wth our eample, we use Taylor seres of U about t = t ad =. However, we ca use our prevous results from the aalyss of the trucato error of spatal dervatves. Specfcally, we have Usg these results, δ 2 U = U U + O( 4 ) δ 2 U = U U + O( 4 )

7 72 τ = U + t U + u [ U + ] [ 6 2 U + O( 4 ) µ U + ] 2 2 U + O( 4 ) The, performg a smlar Taylor seres of the tme dervatve appromato gves, U + U = U t + t 2 tu tt + O( t 2 ). Substtutg ths to τ ad collectg terms powers of t ad gves, τ = U t + u U µu + 2 tu tt + O( t 2 )+ 6 2 u U 2 2 µu + O( 4 ) The frst le of ths equato s actually just the PDE, evaluated at t = t ad =. Sce U s a eact soluto to the PDE, ths s zero. The secod le shows that the tme dscretzato troduces a O( t). The thrd le shows that the spatal dscretzato troduces a O( 2 ) error. Thus, ths umercal method s frst-order accurate tme ad secod-order accurate space. Eercse 3. Dowload the matlab code from Eample. Verfy the ths method s deed secod order accurate space. 50 Upwdg Cosder the oe-dmesoal covecto equato, U t + u U = 0. (08) Recall from Chapter that for the covecto equato, the soluto U(, t) s obtaed by followg the characterstc le back to the tal codto ad evaluatg U o (ξ) where ξ = ut. The covecto equato has some heret drectoalty as the soluto at (,t) depeds oly upo prevous values of U upstream of. As the physcal problem has heret drectoalty, t s atural for our umercal scheme to also have some sort of drectoal bas. Frst cosder the FTCS method from eample. The soluto U + s updated as: U + = U u 2 t (U + U ) Ths s graphcally depcted Fgure 2(a). Ths umercal scheme has o heret drectoalty. Now cosder the Forward Tme-Backward Space (FTBS) method from Eercse 4 whch uses the backwards dfferece formula δ, ad gves the update U + = U u t (U U ). Ths method s depcted graphcally Fgure 2(b). Notce that the umercal scheme s ow based to upstream values. Numercal schemes whch ehbt such a upstream bas are called upwd schemes. Thus, the FTBS method s also kow as the frst-order upwd scheme (sce t s frst-order accurate both space ad tme).

8 73 t t (a) Forward Tme-Cetral Space (b) Forward Tme-Backward Space Fg. 2 Soluto depedece for fte dfferece method. Eercse 4. Dowload the matlab code from Eample ad modfy the code to use the backward dfferece formula δ. Usg u=, t = 000 ad = 50 the FTBS method does ot ehbt the same stablty as the FTCS method. Fg u=, = 50 what s the correspodg value of t above whch the FTBS method beg ehbt stabltes. (a) t = 500 (b) t = 00 (c) t = 50 (d) t = 0 Eercse 5. Fg u=, t = ehbt stabltes. (a) = 500 (b) = 00 (c) = 50 (d) = 0 00 what s the correspodg value of above whch the FTBS method beg Eercse 6. Cosder the frst-order upwd scheme appled to the covecto equato. For what values of t, ad u wll the method retur the eact soluto at the odes? (a) t = 50, = 50, u= (b) t = 50, = 50, u>0 (c) t = u, u>0 (d) both a) ad c)