Accuracy. Computational Fluid Dynamics. Computational Fluid Dynamics. Computational Fluid Dynamics

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1 Computatioal Fluid Dyamics Lecture Jauary 3, 7 Grétar Tryggvaso It is clear that although the umerical solutio is qualitatively similar to the aalytical solutio, there are sigiicat quatitative diereces. The derivatio o the umerical approximatios or the derivatives showed that the error depeds o the size o h ad. First we test or dieret. Evolutio or U=; D=.5; k= N= =.5 m=; time= Number o time steps= T/ Exact Numerical Repeat with a smaller time-step =.5.5 m=; time=.5 Repeat with a smaller time-step =.5.5 m=4; time=.5 N=.5 N= Exact - Exact - Numerical Numerical

2 How accurate solutio ca we obtai? Take =.5 ad N= Very ie spatial resolutio ad a small time step U=; D=.5; k= N= = m=; time=.5 Exact - Numerical Quatiyig the Error Order o Examie the spatial accuracy by takig a very small time step, =.5 ad vary the umber o grid poits, N, used to resolve the spatial directio. The grid size is h = L/N where L = or our case Exact Numerical N E = h ( exact ) = time=.5 N=; E =.633 N =; E =.43 N =4; E =.96 N =6; E =.4 N =8; E =. N =; E =.5 N =; E =. N =6; E = 9.6e-4. Eect o spatial resolutio dt=.5 N= to N=6 - time=.5 I the error is o secod order: E = Ch = C h E - -3 Takig the log: l E = l C h = lc l h -4 / h O a log-log plot, the E versus (/h) curve should thereore have a slope -

3 . Eect o spatial resolutio dt=.5 N= to N=6 E time=.5-4 / h Why is does the error deviate rom the lie or the highest values o N? Trucatio Error Roudo Total Number o Steps, N = /h Summary Fiite dierece approximatios by Taylor expasio Approximatig a partial dieretial equatio Showed, by umerical experimets, that accuracy icreases as the resolutio is icreased Showed that the error behaves i a way that should be predictable As log as accuracy is reasoable, itegratio at larger time steps is more eiciet ad desirable. Ca we icrease the time step ideiitely? Let s repeat the -D advectio-diusio equatio with larges time step. Use =., istead o =.5 Evolutio or U=; D=.5; k= N= =. Exact Numerical

4 Istead o decayig as it should, the amplitude o the umerical solutio keeps icreasig. Ideed, i we cotiued the calculatios, we would evetually produce umbers larger tha the computer ca hadle. This results i a overlow or NaN (Not a Number). Ordiary Dieretial Equatio Take: d dt = The exact solutio is ( t) = e t Forward Euler ODE Example + = with iitial coditio () = + oly i + = ( ) + = ( ) =.5 =.5 =.5 ODE Example + oly i However Backward Euler + = + ODE Example + Obviously, oscillates uless + = ( ) = ( ) = = ( ) or all + = ( + ) Aalysis o the Advectio-Diusio Equatio: Vo Neuma Method

5 Geerally, stability aalysis o the ull oliear system o equatios is too ivolved to be practical, ad we study a model problem that i some way mimics the ull equatios. The liear advectio-diusio equatio is oe such model equatio, ad we will apply vo Neuma's method to check the stability o a simple iite dierece approximatio to that equatio. Cosider the -D advectio-diusio equatio: I iite-dierece orm: + + U = D t x x + U + + = D h h + Look at the evolutio o a small perturbatio = The evolutio o the perturbatio is govered by: + + U + h = D + + h Write the error as a wave (expad as a Fourier series): = ( x ) = Droppig the subscript ikx k e k = ikx = e Recall: e ikx = coskx + isikx The error at ode is: = e ikx + The error at + ad - ca be writte as = e ikx + = e ik (x + h) = e ikx e ikh = e ikx = e ik (x h) = e ikx e ikh Substitutig = e ikx ito yields e ikx e ikx = e ikx e ikh + U + h + U h (eikh e ikx e ikh e ikx ) = D h (eikh e ikx e ikx + e ikh e ikx ) = e ikx e ikh = D + + h + The equatio or the error is: + + U h (eikh e ikh ) = D Solvig or the ratio o the errors: U ikh ikh D = ( e e ) + ( e h h h (eikh + e ikh ) ikh + e ikh )

6 + Dividig by the error amplitude at : Usig: e ikh + e ikh = coskh; ampliicatio actor U ikh ikh D ikh = ( e e ) + ( e h h = U D isikh + (coskh ) h h = 4 D si k h h i U h sikh e ikh e ikh = isikh; + e ikh ) si θ = cosθ + = 4 D h requires that The ratio o the error amplitude at + ad is: si k h i U h sikh + Sice the ampliicatio actor is a complex umber, ad k, the wave umber o the error, ca be aythig, the determiatio o the stability limit is slightly ivolved. We will look at two special cases: (a) U = ad (b) D = (a) Cosider irst the case whe U =, so the problem reduces to a pure diusio + = 4 D h si k h Sice si () the ampliicatio actor is always less tha, ad we id that it is bigger tha - i 4 D h D h (b) Cosider ow the other limit where D = ad we have a pure advectio problem. + = i U h sikh Sice the ampliicatio actor has the orm +i() the absolute value o this complex umber is always larger tha uity ad the method is ucoditioally ustable or this case. + i U h sikh For the geeral case we must ivestigate the stability coditio i more detail. We will ot do so here, but simply quote the results: For a two-dimesioal problem, assume a error o the orm A stability aalysis gives: i, = i(kxi +ly ) e D h ad U D D h 4 ( U + V ) ad 4 D Notice that high velocity ad low viscosity lead to istability accordig to the secod restrictio. For a three-dimesioal problem we get: D h 6 ad ( U + V + W ) 8 D

7 Now you kow! Covergece the solutio to the iite-dierece equatio approaches the true solutio to the PDE havig the same iitial ad boudary coditios as the mesh is reied. Lax s Equivalece Theorem Give a properly posed iitial value problem ad a iitedierece approximatio to it that satisies the cosistecy coditio, stability is the ecessary ad suiciet coditio or covergece. The Modiied Equatio Cosistecy Usig the iite dierece approximatio, we are eectively solvig a equatio that is slightly dieret tha the origial partial dieretial equatios. Does the iite dierece equatio approach the partial dieretial equatio i the limit o zero ad h? Cosistecy Cosider the -D advectio-diusio equatio ad its iite-dierece approximatio + t + U x = D x +U + h = D + + h The discrepacy betwee the two equatios ca be oud by derivig the modiied equatio. Substitutig + Cosistecy + = + = h + h (t) t + (t) t + (x) + 3 (x) h x x = (x) x ito the iite dierece equatio + + U + = D + h + 4 (x) x 4 h + + h t + U x D x Cosistecy Results i the Modiied Equatio: Origial Equatio = (t) t U 3 (x) h x 3 Error terms Shorthad: t + U x D x = O(,h ) 6 + D 4 (x) x 4 h + I this case, the error goes to zero as h ad, so the approximatio is said to be CONSISTENT

8 Cosistecy Although most iite dierece approximatios are cosistet, iocet-lookig modiicatios ca sometimes lead to approximatios that are ot! The Frakel-Duort is a example o a o-cosistet scheme. You will examie it i the homework Cosistecy HW: Examie the Frakel-Duort method Solve the diusio equatio t = D x Usig the Leaprog time itegratio method ad stadard iite-dierece approximatio or the spatial derivative gives: + = D h + + [ ] Leaprog method Now modiy it slightly: Cosistecy + = D h + + [ ] Replace by: = + ( + ) This gives: + = + D h + + ( + ). Which is easily solved or at the ew time step Cosistecy The MODIFIED EQUATION is obtaied by substitutig the expressio or the iite dierece approximatios, icludig the error terms, ito the iite dierece equatio. For a CONSISTENT iite dierece approximatio the error terms go to zero as h ad. The modiied equatio ca ote be used to ier the ature o the error o the iite dierece scheme. More about that later. I the HW, you will examie the error! Multidimesioal Equatios Two-Dimesioal Advectio- Diusio Equatio We will use the model equatio: +U t x + V y = D x + y to demostrate how to solve a partial equatio (iitial value problem) umerically. The extesio to two-dimesios is relatively straight orward, oce the oe-dimesioal problem is ully uderstood.

9 Multidimesioal Equatios For a two-dimesioal low discretize the variables o a two-dimesioal grid + - (x, y) i - i i+ i, + = (x, y + h) i, = (x,y) i +, = (x + h, y) + i, i, Multidimesioal Equatios t + U x + V y = D x The discrete equatio is: = U i+, h D i, i+, + y V i, + h i, + i, i, + i, + h + + i, h i, + Solve or i, or + i, = i, + i, = i, Multidimesioal Equatios + U i+, h D i+, U h : O,h i, + i, V i, + h h + i, + h i, + i, ( i+, i, ) V ( i, + i, ) ( ) + D h A stability aalysis gives: + 4 i, i+, + i, + i, + + ( i, 4 i, ) D h 4 ad ( U + V ) 4 D Example Multidimesioal Equatios t + U x + V y = D x + y + i, = i, U h Multidimesioal Equatios ( i+, i, ) + D h i +, =NY + i, + i, + + ( i, 4 i, ) x = y = =. =. Uiorm low through the domai U=- V= i, stored at each grid poit or give o the boudary =. = = i= i= i=nx

10 Boudary coditios Multidimesioal Equatios Where is give, we simply speciy its value Where the ormal derivative is speciied, we approximate the value at the boudary by oesided diereces At the i= boudary, or example, y = y i, ad by usig i, = h we id that: i, = i, Multidimesioal Equatios % two-dimesioal usteady diusio by the FTCS scheme % =3;m=3;step=;D=.5;legth=.;h=legth/(-); dt=.*.5*h*h/d;=zeros(,m);o=zeros(,m);time=.; u=-.; v=-.; (:,)=.; or l=:step,l,time hold o;mesh(); axis([ m.5]);pause; o=; or i=:-, or =:m- (i,)=o(i,)-(.5*dt*u/h)*(o(i+,)-o(i-,))-... (.5*dt*v/h)*(o(i,+)-o(i,-))+... (D*dt/h^)*(o(i+,)+ o(i,+)+o(i-,)+o(i, -)-4*o(i,)); ed,ed or i=:, (i,)=(i,);ed;or =:m,(,)=(,);(m,)=(m-,);ed; time=time+dt; ed; The usteady evolutio o the solutio Multidimesioal Equatios Multidimesioal Boudary Value Problems (Steady-State) Boudary Value Problems Cosider the Poisso Equatio: + = S x y This equatio has a solutio i or is speciied o the boudary Use stadard iite diereces to discretize: i +, + i, i, + + i, i, + i, = S h h i, For uiorm grids: i+, + i, i, Solve or i, : Boudary Value Problems h ca be writte as + i, + i, + i, h i+, + i, + i, + i, + 4 i, = S h = S i, i, = ( 4 i +, + i, + i, + i, + h S i, )

11 Boudary Value Problems Boudary Value Problems Solve or i, ad use the right had side to compute a ew value. Deote the old values by α ad the ew oes with α+ i, ( ) α + = 4 α i +, + α i, + α i, + α i, + h S i, This iteratio process Jacobi iteratio is very robust but may iteratios are required to reach a accurate solutio. The iteratio must be carried out util the solutio is suicietly accurate. To measure the error, deie the residual: R i, = i+, + i, + i, + i, + 4 i, h S i, At steady-state the residual should be zero. The poitwise residual or the average absolute residual ca be used, depedig o the problem. Ote, simpler criteria, such as the chage rom oe iteratio to the ext is used Boudary Value Problems Although the Jacobi iteratio is a very robust iteratio techique, it coverges VERY slowly. We thereore seek a way to ACCELERATE the covergece to steady-state, makig use o the act that it is oly the steady-state that is o iterest. Here we itroduce the Gauss-Seidler method ad the Successive Over-Relaxatio (SOR) method. Boudary Value Problems The Jacobi iteratio ca be improved somewhat by usig ew values as soo as they become available. + - i- i i+ α + i, = 4 ( α i +, + i, α + α + i, + or =:m or i=: iterate ed ed α + + i, h S i, ) From a programmig poit o view, Gauss-Seidler iteratio is eve simpler tha Jacobi iteratio sice oly oe vector with values is eeded. Boudary Value Problems The Gauss-Seidler iteratio ca be accelerated eve urther by various acceleratio techiques. The simplest oe is the Successive Over-Relaxatio (SOR) iteratio α+ i, = β 4 ( α i+, + α+ α i, + i, + + α+ i, h S i, ) α + ( β) i, Example The SOR iteratio is very simple to program, ust as the Gauss-Seidler iteratio. The user must select the coeiciet. It must be bouded by <β<. β=.5 is usually a good startig value.

12 Boudary Value Problems Boudary Value Problems x + y = = = % two-dimesioal steady-state problem by SOR =4;m=4;iteratios=5;legth=.;h=legth/(-); T=zeros(,m);bb=.7; T(:-,)=.; or l=:iteratios, or i=:-, or =:m- T(i,)=bb*.5*(T(i+,)+... T(i,+)+T(i-,)+T(i,-))+(.-bb)*T(i,); ed,ed % id residual res=; or i=:-, or =:m- res=res+abs(t(i+,)+... T(i,+)+T(i-,)+T(i,-)-4*T(i,))/h^; ed,ed l,res/((m-)*(-)) % Prit iteratio ad residual i (res/((m-)*(-)) <.), break,ed ed; cotour(t); Boudary Value Problems The program is easily modiied or the Jacobi ad the Gauss-Seidler iteratio: Average absolute error:. Number o iteratios Jacobi: 989 Gauss-Seidler: 986 SOR (β =.5): 3 SOR (β =.7): 6 SOR (β =.9): 9 SOR (β =.95): The coverged solutio: 3 Boudary Value Problems 3 4 x + y = Summary smaller time step ad ier resolutio should get us the exact solutio itroduced the vo Neuma method. Fairly mechaical process, we will provide more isight by the iite volume poit o view The modiied equatios helps us see how the approximate ad the exact equatio dier ad i the ormer is cosistet with the latter Multidimesioal advectio-diusio equatio. Essetially the same as the oe-dimesioal problem Iterative methods or boudary value problems. Elemetary approaches to steady state problems