More Elementary Aspects of Numerical Solutions of PDEs!

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1 ttp:// Outlie More Elemetary Aspects o Numerical Solutios o PDEs I tis lecture we cotiue to examie te elemetary aspects o umerical solutios o partial dieretial equatios. Here we will examie: Stability o te liear advectio-diusio equatio Multidimesioal liear advectio-diusio equatio Steady state (boudary value problems) Grétar Tryggvaso Sprig 0 As log as accuracy is reasoable, itegratio at larger time steps is more eiciet ad desirable. Ca we icrease te time step ideiitely? Let s repeat te -D advectio-diusio equatio wit larges time step. Use Δt = 0., istead o Δt = 0.05 Evolutio or U=; D=0.05; k= N= Δt=0. Exact Numerical Istead o decayig as it sould, te amplitude o te umerical solutio keeps icreasig. Ideed, i we cotiued te calculatios, we would evetually produce umbers larger ta te computer ca adle. Tis results i a overlow or NaN (Not a Number). Ordiary Dieretial Equatio

2 Take: d dt = Te exact solutio is ( t) = e t Forward Euler ODE Example + = "t wit iitial coditio (0) = + oly i t " + = ( "t) + = ( "t) Δt=0.5 Δt=.5 Δt=.5 ODE Example + oly i t " However Backward Euler + = + "t + ODE Example + = ( "t) = ( "t) Obviously, oscillates uless = = ( "t) or all Δt t " + = ( + t) Stability Aalysis o te Advectio-Diusio Equatio: Vo Neuma Metod Geerally, stability aalysis o te ull oliear system o equatios is too ivolved to be practical, ad we study a model problem tat i some way mimics te ull equatios. Te liear advectio-diusio equatio is oe suc model equatio, ad we will apply vo Neuma's metod to ceck te stability o a simple iite dierece approximatio to tat equatio. Cosider te -D advectio-diusio equatio: + U = D t x x I iite-dierece orm: U = D " t + Look at te evolutio o a small perturbatio =

3 Te evolutio o te perturbatio is govered by: U = D " t Write te error as a wave (expad as a Fourier series): " ikx k e k = " $ = $ ( x ) = $ Droppig te subscript ikx = e Recall: e ikx = coskx + isikx Te error at ode is: = e ikx + Te error at + ad - ca be writte as = e ikx + = e ik (x + ) = e ikx e ik " = e ikx " = e ik (x ") = e ikx e "ik Substitutig = e ikx ito + " t yields + + e ikx " e ikx t = e ikx e ik + U + " " + U (eik e ikx " e " ik e ikx ) = D (eik e ikx " e ikx + e " ik e ikx ) " = e ikx e "ik = D + " + " + " t Te equatio or te error is: + + U (eik " e " ik ) = D Solvig or te ratio o te errors: " t ik ik D" t = ( e e ) + ( e (eik " + e "ik ) U ik ik + e ) + Dividig by te error amplitude at : Usig: e ik + e ik = cosk; ampliicatio actor " t ik ik D" t = ( e e ) + ( e = U"t D"t isik + (cosk ) U ik ik = 4 D"t si k i U"t sik e ik e ik = isik; + e ) si =" cos Te ratio o te error amplitude at + ad is: + =" 4 Dt si k " i Ut sik Stability requires tat + " Sice te ampliicatio actor is a complex umber, ad k, te wave umber o te error, ca be aytig, te determiatio o te stability limit is sligtly ivolved. We will look at two special cases: (a) U = 0 ad (b) D = 0

4 (a) Cosider irst te case we U = 0, so te problem reduces to a pure diusio + = " 4 Dt si k Sice si () te ampliicatio actor is always less ta, ad we id tat it is bigger ta - i " 4 Dt " Dt " (b) Cosider ow te oter limit were D = 0 ad we ave a pure advectio problem. + =" i Ut sik Sice te ampliicatio actor as te orm +i() te absolute value o tis complex umber is always larger ta uity ad te metod is ucoditioally ustable or tis case. + i Ut sik For te geeral case we must ivestigate te stability coditio i more detail. We will ot do so ere, but simply quote te results: For a two-dimesioal problem, assume a error o te orm A stability aalysis gives: i, = i(kxi +ly ) e td " ad U t D " D" t 4 ( U + V ) " t ad 4 D Notice tat ig velocity ad low viscosity lead to istability accordig to te secod restrictio. For a tree-dimesioal problem we get: Dt " 6 ad ( U + V + W ) t " 8 D Stability Now you kow Covergece te solutio to te iite-dierece equatio approaces te true solutio to te PDE avig te same iitial ad boudary coditios as te mes is reied. Lax s Equivalece Teorem Give a properly posed iitial value problem ad a iitedierece approximatio to it tat satisies te cosistecy coditio, stability is te ecessary ad suiciet coditio or covergece. Summary Itroduced a ormal metod to examie weter a give iite dierece approximatio is stable or ot Itroduced Lax s equivalet teorem.

5 Multidimesioal Equatios Two-Dimesioal Advectio- Diusio Equatio We will use te model equatio: +U t x + V y = D " $ x + % y & to demostrate ow to solve a partial equatio (iitial value problem) umerically. Te extesio to two-dimesios is relatively straigt orward, oce te oe-dimesioal problem is ully uderstood. Multidimesioal Equatios For a two-dimesioal low discretize te variables o a two-dimesioal grid + - (x, y) i - i i+ i, + = (x, y + ) i, = (x,y) i +, = (x +, y) + i, i, Δt Multidimesioal Equatios t + U x + V y = D x Te discrete equatio is: = U i+, D i, i+, + y V i, + i, + i, i, + i, i, i, + Solve or i, or + i, = i, + i, = i, Multidimesioal Equatios + Δt U i+, D i+, ΔtU i, + i, V i, + + i, + i, + i, ( i+, i, ) ΔtV ( i, + i, ) ( ) Accuracy: O t, + ΔtD A stability aalysis gives: + 4 i, i+, + i, + i, + + ( i, 4 i, ) td " 4 ad ( U + V ) t " 4 D Example

6 Multidimesioal Equatios t + U x + V y = D x + y + i, = i, "tu Multidimesioal Equatios ( i+, i, ) + "td i +, =NY + i, + i, + + ( i, 4 i, ) x = 0 y = 0 =0.0 =.0 Uiorm low troug te domai U=- V=0 i, stored at eac grid poit or give o te boudary =0.0 = = i= i= i=nx Boudary coditios Multidimesioal Equatios Were is give, we simply speciy its value Were te ormal derivative is speciied, we approximate te value at te boudary by oesided diereces At te i= boudary, or example, y = 0 y i, ad by usig i, = 0 we id tat: i, = i, Multidimesioal Equatios % two-dimesioal usteady diusio by te FTCS sceme % =3;m=3;step=0;D=0.05;legt=.0;=legt/(-); dt=.0*0.5**/d;=zeros(,m);o=zeros(,m);time=0.0; u=-0.0; v=-.0; (:,)=.0; or l=:step,l,time old o;mes(); axis([0 0 m 0.5]);pause; o=; or i=:-, or =:m- (i,)=o(i,)-(0.5*dt*u/)*(o(i+,)-o(i-,))-... (0.5*dt*v/)*(o(i,+)-o(i,-))+... (D*dt/^)*(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; Te usteady evolutio o te solutio Multidimesioal Equatios Multidimesioal Boudary Value Problems (Steady-State)

7 Boudary Value Problems Cosider te Poisso Equatio: + = S x y Tis equatio as a solutio i or is speciied o te boudary Use stadard iite diereces to discretize: i +, + i, i, + + i, i, + i, = S i, For uiorm grids: i+, + i, i, ca be writte as Solve or i, : Boudary Value Problems + i, + i, + i, i+, + i, + i, + i, + 4 i, = S = S i, i, = ( i +, i, i, i, + S i, ) Boudary Value Problems Boudary Value Problems Solve or i, ad use te rigt ad side to compute a ew value. Deote te old values by α ad te ew oes wit α+ i, ( ) + = 4 i +, + i", + i, " + i, + " S i, Tis iteratio process Jacobi iteratio is very robust but may iteratios are required to reac a accurate solutio. Te iteratio must be carried out util te solutio is suicietly accurate. To measure te error, deie te residual: R i, = i+, + i, + i, + i, + 4 i, S i, At steady-state te residual sould be zero. Te poitwise residual or te average absolute residual ca be used, depedig o te problem. Ote, simpler criteria, suc as te cage rom oe iteratio to te ext is used Boudary Value Problems Altoug te Jacobi iteratio is a very robust iteratio tecique, it coverges VERY slowly. We tereore seek a way to ACCELERATE te covergece to steady-state, makig use o te act tat it is oly te steady-state tat is o iterest. Here we itroduce te Gauss-Seidler metod ad te Successive Over-Relaxatio (SOR) metod. Boudary Value Problems Te Jacobi iteratio ca be improved somewat by usig ew values as soo as tey become available. + - i- i i+ + i, = 4 ( i +, + i", + + i, + or =:m or i=: iterate ed ed + + i, " " S i, ) From a programmig poit o view, Gauss-Seidler iteratio is eve simpler ta Jacobi iteratio sice oly oe vector wit values is eeded.

8 Boudary Value Problems Te Gauss-Seidler iteratio ca be accelerated eve urter by various acceleratio teciques. Te simplest oe is te Successive Over-Relaxatio (SOR) iteratio + i, = " 4 ( i +, + i, + ( ") i, + + i, i, + S i, ) Example Te SOR iteratio is very simple to program, ust as te Gauss-Seidler iteratio. Te user must select te coeiciet. It must be bouded by <β<. β=.5 is usually a good startig value. Boudary Value Problems Boudary Value Problems x + y = 0 = = % two-dimesioal steady-state problem by SOR =40;m=40;iteratios=5000;legt=.0;=legt/(-); T=zeros(,m);bb=.7; T(0:-0,)=.0; or l=:iteratios, or i=:-, or =:m- T(i,)=bb*0.5*(T(i+,)+... T(i,+)+T(i-,)+T(i,-))+(.0-bb)*T(i,); ed,ed % id residual res=0; or i=:-, or =:m- res=res+abs(t(i+,)+... T(i,+)+T(i-,)+T(i,-)-4*T(i,))/^; ed,ed l,res/((m-)*(-)) % Prit iteratio ad residual i (res/((m-)*(-)) < 0.00), break,ed ed; cotour(t); Boudary Value Problems Te program is easily modiied or te Jacobi ad te Gauss-Seidler iteratio: Average absolute error: 0.00 Number o iteratios Jacobi: 989 Gauss-Seidler: 986 SOR (β =.5): 30 SOR (β =.7): 6 SOR (β =.9): 9 SOR (β =.95): Te coverged solutio: Boudary Value Problems x + y =

9 Summary Stability itroduced te vo Neuma metod. Fairly mecaical process, we will provide more isigt by te iite volume poit o view Multidimesioal advectio-diusio equatio. Essetially te same as te oe-dimesioal problem Iterative metods or boudary value problems. Elemetary approaces to steady state problems