INF5620 Key exercise 1

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1 INF6 Key exerise Per Kristia Igebrigtse Program ode ad aimated plots fod at Sprig

2 INF6 Key exerise Per Kristia Igebrigtse Cotets The D wave eqatio with dampig. Nmerial sheme Bodary oditios Ope bodary at x = L Iitial step Stability Reslts Notes/sage Plots Pytho ode The D wave eqatio with dampig. Nmerial sheme The D wave eqatio with dampig with variable wave veloity is t + β t = + x x y y where β is a dampig fator ad = x, y is the wave veloity. The partial derivatives will be replaed by etered differees throghot. Usig the otatio from the letres, the wave eqatio beomes [ Dt D t + βd t = D x D x + D y D y ] where i ad j are idies for x ad y respetively. Isertig the etered differees diretly, we get + + t Now we may solve for β + + β t t + Gropig ad reshfflig terms: + +β t = + ad fially = D x i+/,j i /,j x + D y +/ / y = x i+/,j i+,j i /,j i,j + y +/ + / = t x i+/,j i+,j i /,j i,j β t + t + t y +/ + + = +β t + t y +/ + / i+,j i /,j i,j x i+/,j / [ + β t + t x i+/,j i+,j + t y +/ i /,j i,j + / ]

3 INF6 Key exerise Per Kristia Igebrigtse I this implemetatio, will oly be kow at the grid poits, so the vales betwee grid poits eed to be replaed. Choosig a simple average gives i+/,j = i+,j + i /,j = + i,j +/ = i+,j + / = + givig the fll sheme for the ier poits + = +β t [ + β t + t x i+,j + i+,j + i,j + t y + + i,j + + ]. Bodary oditios At the bodaries x =,x = L, y =,y = L, we reqire that the derivatives x ad y vaish. Stayig osistet, we se etered differees for these derivatives at the bodaries. Treatig oly the bodary at x = : x =,j,j = x,j =,j The other three bodaries get eqivalet oditios: i, = i, N x+,j = N x,j i,n y+ = i,n y givig s relatios for the ghost poits otside the domai. By evalatig the sheme at i = ad sbstittig for the ghost poit at i = whih is eqivalet to swithig the idex i withi + we get the fll bodary oditio at x = : +,j = +β t [,j +,j β t + t x,j +,j,j,j,j +,j + t y,j+ +,j,j,j,j+,j,j +,j,j ],j Eqivalet expressios omitted arise for the other bodaries ad orer poits sig the same method. 3 3

4 INF6 Key exerise Per Kristia Igebrigtse Ope bodary at x = L It is possible to let the wave travel ot of the domai i oe diretio, say at the bodary x = L. Here the bodary oditio reads t = x By sig etered differees for the partial derivatives, we get the approximatio at i = Nx: + Nx,j Nx,j Nx+,j = Nx,j Nx,j t x Sie this is a bodary oditio, we eed to solve this for Nx+,j to aot for the ghost poit i the fll sheme: x Nx,j t + Nx,j Nx,j = Nx+,j Nx,j Nx+,j = Nx,j x Nx,j t + Nx,j Nx,j This a be iserted ito the fll sheme at i = N x, whih may the be solved for + Nx,j, ad we get a ope bodary oditio at x = L. This has bee implemeted i the aompayig sore ode..3 Iitial step To allate the vale of + at a ew timestep +, the sheme derived i setio. reqires steps ad to be kow. The first timestep,however,reqires ad, the latter of whih is a ghost poit before the iitial oditio. Examiig the iitial oditio for, wehave { os πx for x< x, y, = otherwise whih meas that t= =. Usig a etered differee for this other iitial oditio, we get t = t = Whe we make this sbstittio i the sheme, we get for the iitial step = [ +β t + β t whih leads to + t x i+,j + i+,j + i,j + t y + + i,j + + ] β t/ β t/+ =+β t/... = β t/+ + β t/... =... = + t 4 x i+,j + i+,j + i,j i,j + t 4 y The appropriate adjstmets for bodaries ad orers is doe i the same way as i setio.. 4

5 INF6 Key exerise Per Kristia Igebrigtse Stability Derivig stability riteria for the D wave eqatio with variable wave veloity ad dampig is omplex ad time-osmig. Aordig to eqatio.66 i HPL hapter.4 page 63 the stability riterio for variable wave veloity o dampig is t max x, y / / + 4 x y A attempt at isertig a trial soltio A exp i k x î x + k y j y ω t ito the fll differee eqatio for or problem revealed that the dampig gives rise to imagiary terms. There was isffiiet time to do frther work o this, so eqatio 4 has bee sed for the atal implemetatio.. Reslts A seletio of plots for two differet iitial oditios are ilded i setio.7. All plots are reated with the followig settigs L = N x = 6 N y = 6 ier =. oter =. Tmax =. A variety of aimated plots a be fod at alog with the fll sore ode also ilded i this report. The soltio seems well-behaved for the hose spatial ad temporal grid ad other parameters. The osie wave leaves some osillatios i its wake, whih I believe are de to the shape of the wave, as the osie wave has a disotiity. Aordig to the iitial oditio, the wave eds abrptly at x = /. If we assme that the srfae arod the wave is flat, x will be zero as we approah x =/from the positive side. However, if we approah x =/ from the egative side, x x / = x os πx x /. These osillatios are ot apparet for the expoetial rve. We a also learly see the effet of the dampig, whe omparig damped ad damped wave plots side-by-side..6 Notes/sage The spatial poit loop has bee implemeted sig vetorized expressios, as I feel omfortable sig them. The ode opes for adjstig several parameters from the ommad lie. Usage: $ pytho wave.py [IC] [BC] [Parameter list param=vale] It ildes for differet iitial oditios: zero The othig shold happe soltio, = os Cosie wave expwave Bell rve waveform startig at expdrop D Bell rve etered i domai The bodary oditio a be either fixed free Fixed bodaries = at bodaries Free bodaries d/d = at bodaries

6 INF6 Key exerise Per Kristia Igebrigtse ad fially there is a list of parameters. Lx x size of domai Ly y size of domai Nx Nmber of grid poits i x Ny Nmber of grid poits i y oter Wave veloity i oter domai ier Wave veloity i ier domai Tmax Max/ed time B Dampig oeffiiet bxl=\ ope\ Allows for ope bodary at x = L, ie. the wave may travel ot of the domai 6

7 7.7 Plots Time.484 Time Time Figre : I x, y = os πx for x</ Ier domai =., oter domai =., o dampig Time Figre : I x, y = os πx for x</ Ier domai =., oter domai =., dampig B = Time.468 Time INF6 Key exerise Per Kristia Igebrigtse perki@stdet.matat.io.o

8 Time Time Time Figre 3: I x, y =e x Ier domai =., oter domai =., o dampig Time Figre 4: I x, y =e x Ier domai =., oter domai =., dampig B = Time.8338 Time INF6 Key exerise Per Kristia Igebrigtse perki@stdet.matat.io.o

9 9.8 Pytho ode from sitools.std import def solverxv,yv,lx,ly,dx,dx,dy,dy,nx,ny,dt,dt,tmax,,,,,,b,b= Free,bxL=Noe: # dampig oeffiiet sed i allatios b = B dt. # x = bodary [,:Ny] = [,:Ny] +. dt/dx [,:Ny] + [,:Ny] [,:Ny] [,:Ny] \ [,:Ny] + [,:Ny] [,:Ny] [,:Ny] \ +. dt/dy [,:Ny+] + [,:Ny] [,:Ny+] \ [,:Ny] [,:Ny] + [,:Ny ] [,:Ny] [,:Ny ] # x = L bodary [Nx,:Ny] = [Nx,:Ny] +. dt/dx [Nx,:Ny] + [Nx,:Ny] [Nx,:Ny] [Nx,:Ny] \ [Nx,:Ny] + [Nx,:Ny] [Nx,:Ny] [Nx,:Ny] \ +. dt/dy [Nx,:Ny+] + [Nx,:Ny] [Nx,:Ny+] \ [Nx,:Ny] [Nx,:Ny] + [Nx,:Ny ] [Nx,:Ny] [Nx,:Ny ] # y = bodary [:Nx,] = [:Nx,] +. dt/dx [:Nx+,] + [:Nx,] [:Nx+,] [:Nx,] \ [:Nx,] + [:Nx,] [:Nx,] [:Nx,] \ +. dt/dy [:Nx,] + [:Nx,] [:Nx,] \ [:Nx,] [:Nx,] + [:Nx,] [:Nx,] [:Nx,] # y = L bodary [:Nx,Ny] = [:Nx,Ny] +. dt/dx [:Nx+,Ny] + [:Nx,] [:Nx+,Ny] [:Nx,Ny] \ [:Nx,Ny] + [:Nx,Ny] [:Nx,Ny] [:Nx,Ny] \ +. dt/dy [:Nx,Ny ] + [ : Nx,Ny] [:Nx,Ny ] \ [:Nx,Ny] [:Nx,Ny] + [:Nx,Ny ] [:Nx,Ny] [:Nx,Ny ] #!!! do orer poits here!!! [,] = [,] +. dt/dx [,] + [,] [,] [,] \ [,] + [,] [,] [,] \ +. dt/dy [,] + [,] [,] \ [,] [,] + [,] [,] [,] [Nx,] = [Nx,] +. dt/dx [Nx,] + [Nx,] [Nx,] [Nx,] \ [Nx,] + [Nx,] [Nx,] [Nx,] \ +. dt/dy [Nx,] + [Nx,] [Nx,] [Nx,]\ [Nx,] + [Nx,] [Nx,] [Nx,] [,Ny] = [,Ny] +. dt/dx [,Ny] + [,Ny] [,Ny] [,Ny] \ [,Ny] + [,Ny] [,Ny] [,Ny] \ +. dt/dy [,Ny ] + [,Ny] [,Ny ] [,Ny] \ [,Ny] + [,Ny ] [,Ny] [,Ny ] [Nx,Ny] = [Nx,Ny] +. dt/dx [Nx,Ny] + [Nx,Ny] [Nx,Ny] [Nx,Ny] \ [Nx,Ny] + [Nx,Ny] [Nx,Ny] [Nx,Ny] \ +. dt/dy [Nx,Ny ] + [Nx,Ny] [Nx,Ny ] [Nx,Ny] \ [Nx,Ny] + [Nx,Ny ] [Nx,Ny] [Nx,Ny ] # ier poits [:Nx,:Ny] = [:Nx,:Ny] \ +. dt/dx [:Nx+,:Ny] + [:Nx,:Ny] [:Nx+,:Ny] [:Nx,:Ny] \ [:Nx,:Ny] + [:Nx,:Ny] [:Nx,:Ny] [:Nx,:Ny] \ +. dt/dy [:Nx,:Ny+] + [:Nx,:Ny] [:Nx,:Ny+] [:Nx,:Ny] \ [:Nx,:Ny] + [:Nx,:Ny ] [:Nx,:Ny] [:Nx,:Ny ] mesh xv, yv,,title= Time %f % f l o a t., zlim =[.,.] #mesh xv, yv,,title= Time %f % float.,zlim=[.,.], hardopy= wave. pg mesh xv, yv,,title= Time %f % f l o a t dt, zlim =[.,.] #mesh xv, yv,,title= Time %f % floatdt,zlim=[.,.], hardopy= wave. pg = # r mai time loop for t i rage, it Tmax/dt+: INF6 Key exerise Per Kristia Igebrigtse perki@stdet.matat.io.o

10 if b== free : # x = bodary [,:Ny] =.+b. [,:Ny] + [,:Ny] b. +. dt/dx [,:Ny] + [,:Ny] [,:Ny] [,:Ny] \ [,:Ny] + [,:Ny] [,:Ny] [,:Ny] \ +. dt/dy [,:Ny+] + [,:Ny] [,:Ny+] \ [,:Ny] [,:Ny] + [,:Ny ] [,:Ny] [,:Ny ] # x = L bodary if bxl== ope : [Nx,:Ny] =. + dt [Nx,:Ny] \ + [Nx,:Ny]/. [Nx,:Ny] dx.+b. [Nx,:Ny] \ + [Nx,:Ny] b. \ +. dt/dx [Nx,:Ny] + [Nx,:Ny] [Nx,:Ny] \ + dx/ [Nx,:Ny] dt [Nx,:Ny] [Nx,:Ny] \ [Nx,:Ny] + [Nx,:Ny] [Nx,:Ny] [Nx,:Ny] \ +. dt/dy [Nx,:Ny+] + [Nx,:Ny] [Nx,:Ny+] \ [Nx,:Ny] [Nx,:Ny] + [Nx,:Ny ] [Nx,:Ny] [Nx,:Ny ] else : [Nx,:Ny] =.+b. [Nx,:Ny] + [Nx,:Ny] b. \ +. dt/dx [Nx,:Ny] + [Nx,:Ny] [Nx,:Ny] [Nx,:Ny] \ [Nx,:Ny] + [Nx,:Ny] [Nx,:Ny] [Nx,:Ny] \ +. dt/dy [Nx,:Ny+] + [Nx,:Ny] [Nx,:Ny+] \ [Nx,:Ny] [Nx,:Ny] + [Nx,:Ny ] [Nx,:Ny] [Nx,:Ny ] # y = bodary [:Nx,] =.+b. [:Nx,] + [:Nx,] b. \ +. dt/dx [:Nx+,] + [:Nx,] [:Nx+,] [:Nx,] \ [:Nx,] + [:Nx,] [:Nx,] [:Nx,] \ +. dt/dy [:Nx,] + [:Nx,] [:Nx,] \ [:Nx,] [:Nx,] + [:Nx,] [:Nx,] [:Nx,] # y = L bodary [:Nx,Ny] =.+b. [:Nx,Ny] + [:Nx,Ny] b. \ +. dt/dx [:Nx+,Ny] + [:Nx,Ny] [:Nx+,Ny] [:Nx,Ny] \ [:Nx,Ny] + [:Nx,Ny] [:Nx,Ny] [:Nx,Ny] \ +. dt/dy [:Nx,Ny ] + [ : Nx,Ny] [:Nx,Ny ] \ [:Nx,Ny] [:Nx,Ny] + [:Nx,Ny ] [:Nx,Ny] [:Nx,Ny ] #!!! do orer poits here!!! [,] =.+b. [,] + [,] b. \ +. dt/dx [,] + [,] [,] [,] \ [,] + [,] [,] [,] \ +. dt/dy [,] + [,] [,] [,] [,] + [,] [,] [,] if bxl== ope : [Nx,] =. + dt [Nx,] \ + [Nx,]/. [Nx,] dx.+b. [Nx,] + [Nx,] b. \ +. dt/dx [Nx,] + [Nx,] [Nx,] + dx/[nx,] dt [Nx,] \ [Nx,] [Nx,] + [Nx,] [Nx,] [Nx,] \ +. dt/dy [Nx,] + [Nx,] [Nx,] [Nx,] \ [Nx,] + [Nx,] [Nx,] [Nx,] [Nx,Ny] =. + dt [Nx,Ny] \ + [Nx,Ny]/. [Nx,Ny] dx.+b. [Nx,Ny] + [Nx,Ny] b. \ +. dt/dx [Nx,Ny] + [Nx,Ny] [Nx,Ny] + dx/ [Nx,Ny] dt [Nx,Ny] \ [Nx,Ny] [Nx,Ny] + [Nx,Ny] [Nx,Ny] [Nx,Ny] \ +. dt/dy [Nx,Ny ] + [Nx,Ny] [Nx,Ny ] [Nx,Ny] \ [Nx,Ny] + [Nx,Ny ] [Nx,Ny] [Nx,Ny ] else : [Nx,] =.+b. [Nx,] + [Nx,] b. \ +. dt/dx [Nx,] + [Nx,] [Nx,] \ [Nx,] [Nx,] + [Nx,] [Nx,] [Nx,] \ +. dt/dy [Nx,] + [Nx,] [Nx,] [Nx,] \ [Nx,] + [Nx,] [Nx,] [Nx,] [Nx,Ny] =.+b. [Nx,Ny] + [Nx,Ny] b. \ +. dt/dx [Nx,Ny] + [Nx,Ny] [Nx,Ny] [Nx,Ny] \ [Nx,Ny] + [Nx,Ny] [Nx,Ny] [Nx,Ny] \ +. dt/dy [Nx,Ny ] + [Nx,Ny] [Nx,Ny ] [Nx,Ny] \ [Nx,Ny] + [Nx,Ny ] [Nx,Ny] [Nx,Ny ] [,Ny] =.+b. [,Ny] + [,Ny] b. \ INF6 Key exerise Per Kristia Igebrigtse perki@stdet.matat.io.o

11 +. dt/dx [,Ny] + [,Ny] [,Ny] [,Ny] \ [,Ny] + [,Ny] [,Ny] [,Ny] \ +. dt/dy [,Ny ] + [,Ny] [,Ny ] [,Ny] \ [,Ny] + [,Ny ] [,Ny] [,Ny ] # ier poits [:Nx,:Ny] =.+b. [:Nx,:Ny] + [:Nx,:Ny] b. \ +. dt/dx [:Nx+,:Ny] + [:Nx,:Ny] [:Nx+,:Ny] [:Nx,:Ny] \ [:Nx,:Ny] + [:Nx,:Ny] [:Nx,:Ny] [:Nx,:Ny] \ +. dt/dy [:Nx,:Ny+] + [:Nx,:Ny] [:Nx,:Ny+] [:Nx,:Ny] \ [:Nx,:Ny] + [:Nx,:Ny ] [:Nx,:Ny] [:Nx,:Ny ] mesh xv, yv,, t i t l e= Time %f % f l o a t t dt, zlim =[.,.], olorbar= o #mesh xv, yv,, t i t l e = Time %f % f l o a t t dt, zlim =[.,.], olorbar= o,show=false, hardopy= expwave d %4d. ps % = + [:,:] = [:,:] = # movie oswave d.pg, eoder= overt, fps=float/tmax, otpt file= oswave d.gif def r i i ti al, b,lx=,ly=,nx=6,ny=6, oter=., ier=.,tmax=.,b=.,bxl=noe: # set p spatial grid poits xv = lispae,lx,nx+ yv = lispae,ly,nx+ dx = xv [ ] xv [ ] dy = yv [ ] yv [ ] dx = dx dx dy = dy dy # wave veloity = zeros Nx+, Ny+ = zeros Nx+, Ny+ [:,:] = oter [Nx/ 8:Nx/+8,Ny/ 8:Ny/+8] = ier [ :, : ] = [:,:] # time step size dt =.9./.max./ sqrt./dx +./dy dt = dt dt # soltio arrays = zeros Nx+,Ny+ = zeros Nx+,Ny+ = zeros Nx+,Ny+ # i i t i a l oditio def Ix,y,d: if d == expwave : retr exp x elif d == expdrop : retr. exp x Lx /. /. y Ly /. /. elif d == os : retr os pi x if x <. else. elif d == zero : retr. # apply i i t i a l oditio for i i rage,nx+: for j i rage,ny+: [i ][j] = Idx i,dx j,iitial # solve it! solverxv,yv,lx,ly,dx,dx,dy,dy,nx,ny,dt,dt,tmax,,,,,,b,b,bxl INF6 Key exerise Per Kristia Igebrigtse perki@stdet.matat.io.o

12 if ame == mai : import sys if lesys. argv < : r i i t i a l= expdrop,b= free, ier=.,tmax=.,b=. elif lesys. argv == : md = r i i t i a l =\ %s \, b=\ free\ % sys. argv [] else : md = r i i t i a l =\ %s \, b=\ %s \, %s % s y s. argv [ ], s y s. argv [ ],,.joisys.argv[3:] prit md evalmd INF6 Key exerise Per Kristia Igebrigtse perki@stdet.matat.io.o