HW #4 Solutions: M697J Fall 2006

Transcription

1 HW #4 Solutios: M697J Fall (4.2 BHM) Eigevalue calculatios i two dimesios. Cosider the weighted Jacobi method applied to the model Poisso equatio i two dimesios o the uit square. Assume a uiform grid with h = i each directio. (a) Let v ij be the approximatio to the solutio at the grid poit (x i,y j ). Write the (i,j)th equatio of the correspodig discrete problem, where i,j. (b) Lettig A be the matrix of coefficiets for the discrete system, write the (i,j)th equatio for the eigevalue problem Av = λv. (c) Assume ad eigevector solutio of the form ( ) ikπ v ij = si si ( jlπ ), k,l. Usig sie additio rules, simplify this eigevalue equatio, cacel commo terms, ad show that the eigevalues are [ ( ) ( )] kπ lπ λ kl = 4 si 2 + si 2, k,l. 2 2 (d) As i the oe-dimesioal case, ote that the iteratio matrix of the weighted Jacobi method is give by P ω = I ωd A, where D correspods to the diagoal terms of A. Fid the eigevalues of P ω. (e) Usig a graphig utility, fid a suitable way to preset the two-dimesioal set of eigevalues (either a surface plot or multiple curves). Plot the eigevalues for ω = 2/3,4/5, ad = 6. (f) I each case, discuss the effect of the weighted Jacobi method o the low- ad high-frequecy modes. Be sure to ote that modes ca have high frequecies i oe directio ad low frequecies i the other directio. (g) What do you coclude about the optimal value of ω for the two-dimesioal problem? ANS: (a) We approximate the model problem u = f, u Γ = 0 by Dx 2v D2 yv = f, or v i,j + 2v i,j v i+,j h 2 + v i,j + 2v i,j v i,j+ h 2 = f i,j. (b) Note: give that there is o divisio by h 2 i the defitio of λ kl i part (c) it is ot icluded here. It may be that BHM meat the homogeeous problem i part (a). I either case, we have ( v i,j + 2v i,j v i+,j ) + ( v i,j + 2v i,j v i,j+ ) = λv i,j. (c) Due to liearity we ca treat each term o the left i (b) separately. The oe has ( ( ) v i,j + 2v i,j v i+,j = si (i )kπ + 2si ( ) ( )) ikπ si (i+)kπ si ( jlπ ).

2 Note that With this, ( ) (i ± )kπ si = si v i,j + 2v i,j v i+,j ( ) ikπ cos = ( 2cos ( kπ ( ) kπ ± cos ( ) ikπ si ( ) kπ. ) ( si ikπ ) ( + 2si ikπ )) ( ) si jlπ = 2 ( 2cos ( )) ( kπ si ikπ ) ( ) si jlπ = 4si 2 ( ) ( kπ 2 si ikπ ) ( ) si jlπ = λ x si ( ikπ ) si ( jlπ ), or, λ x = 4si 2 ( kπ 2). Repeatig these steps for the vi,j + 2v i,j v i,j+ term, ad addig, gives the result. (d) Agai, due to liearity, the kl th eigevalue of P ω is (P ω ) kl = (I ωd A) kl = ω [ ( ) kπ 4 λ kl = ω si 2 2 (e) (images o ext page!) ( )] lπ + si 2 2 for k,l.

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4 2. (4.5 BHM) Two-dimesioal program. For the two-dimesioal problem, proceed agai i a modular way: (a) Write a subroutie that performs weighted Jacobi o a two-dimesioal grid. Withi the subroutie, it is easiest to refer to v ad f as two-dimesioal arrays. (b) Make the appropriate modificatios to the oe-dimesioal code to implemet biliear iterpolatio ad full weightig o a two-dimesioal grid. (c) Make the (mior) chages required i the mai program to create a two-dimesioal V-cycle program. Test this program o problems with kow exact solutios. For example, for fixed k ad l, take f(x,y) = C si (kπx) si (lπy) o the uit square (0 x,y ) where C is a costat. The C u(x,y) = π 2 k 2 + π 2 l 2 si(kπx) si (lπy). + σ (c) Preset results for k = 0, l = 20 takig M = N = 32. Use the optimal Jacobi weight ω = 4/5. (d) With your code ad the Jacobi weight ω = 4/5, reproduce the = 64 portio of the table o page 65 of the coursebook, makig sure to use the discrete L 2 orm h. Recall for d-dimesios, if h = h x = h x2 = = h xd the ( u h h = h d i ) {u h i } 2 /2 ( ) = h d/2 {u h i } 2 /2 = h d/2 u h 2. So we see, i this case the discrete L 2 orm is simply the stadard 2-orm, 2, scaled. Also ote that you ca use Matlab s orm to compute u h 2. ANS: The V-cycle code is available o the course website. Here are the results for part (d): cycle(s) res rat err rat e e e e e e e e e e e e e e e e e e e e-06 i

5 3. Cosider the 2-poit oe-dimesioal BVP { u + u = (π 2 si πx 2π cos πx)e x u(0) = u() = 0. (a) Write a MATLAB script to solve the problem by the FFT method, usig the Discrete Sie Trasform as implemeted by dst.m applied to the 2d order cetered FD scheme, assumig σ 0 is a costat, D 2 v i + σv i = f i, where D 2 = D + D. Assume a meshsize h = /2 p, where p is a positive iteger. For p = : 4, plot the exact solutio (u(x) vs. x) ad the umerical solutio (v i vs. x i ), icludig the boudary poits. The 4 plots should appear separately i oe figure, with axes labeled ad a title for each idicatig p. Ivestigate subplot i MATLAB for how to have multiple plots i a sigle figure widow. (b) For p = : 5 preset a table with the followig data - colum : h; colum 2: u h v h ; colum 3: u h v h /h 2 ; colum 4: cpu time; colum 5: (cpu time)/( log ), where h = /. Discuss the treds i each colum. Iclude a copy of your code. ANS: Here is the code for the D solver: fuctio [u,x] = poissd_fft(n,sigma,f_fuc) Poisso solve usig DST applied to: -u + \sigma u = f, u(0)=u()=0 h = /N; x = (0:N) *h; lam = 2*(-cos(x(2:N)*pi))/(h^2); f = eval(f_fuc); u = dst(f(2:n)); trasform u = u./(lam+sigma); Solve i Fourier space u = dst(u); trasform back u = [0; u; 0]; set BC Here is the code for the computatios ad plots, followed by the result: xx = 0:0.0:;xx=xx ; uu = si(pi*xx).*exp(xx); sigma = ; f = (pi^2*si(pi*x)-2*pi*cos(pi*x)).*exp(x) ; clf; for p=:4 [v,x] = poissd_fft(2^p,sigma,f); subplot(2,2,p),plot(xx,uu,x,v, * ),grid leged( u, v,2) axis( tight ),xlabel( x ),title([ p=,um2str(p)]) ed

6 p= p=2.5 u v.5 u v x p= x p=4.5 u v.5 u v x x (b) Here is the code. Note that the solver is called 50 times ad the cpu time averaged i order to get a accurate timig. sigma = ; f = (pi^2*si(pi*x)-2*pi*cos(pi*x)).*exp(x) ; h = zeros(5,); = zeros(5,); times = zeros(5,); err_if = zeros(5,); for p = :5 p tic; for j=:50 [v,x] = poissd_fft(2^p,sigma,f); ed stime = toc; h(p) = /(2^p); (p) = 2^p; times(p) = stime/50; u = si(pi*x).*exp(x); err_if(p) = max(abs(u-v)); ed format short e disp( ) disp( h if_err err/h^2 cputime cpu/(log) ) disp( ) disp( ) disp([h err_if err_if./(h.^2) times times./(.*log2())])

7 Here are the results. I executed the code twice so that FFTW was properly iitialized for the vector legths required. h if_err err/h^2 cputime cpu/(log) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-0.996e e e e e e e e e e e e e e e e e e e e e e-07 We ca see from the err/h 2 colum that the expected O(h 2 ) error is observed util h.2207e 04 sice util this poit err/h 2 approaches a costat. But the we lose accuracy sice roudoff error begis to domiate. Thus, while theoretically as h 0 we have covergece, floatig poit errors i the form of roudoff error evetually domiates! Note, that the cputime does ot exhibit the O( log ) scalig util is large. This is due to the fact that there are also O() computatios which are a sigificat portio of the total time util is large.

8 4. Cosider the 2D BVP o [0,] [0,] { u + σu = f(x,y) u Γ = g(x,y). (a) Write a MATLAB script to solve the problem by the FFT method, usig the Discrete Sie Trasform as implemeted by dst.m applied to the 2d order cetered FD scheme, assumig σ 0 is a costat, (D 2 x + D2 y )v i,j + σv i,j = f i,j, where D 2 = D + D. Your code should be geeral eough to hadle ihomogeeous BC (g 0) ad M N, i.e., /M = h x h y = /N. (b) Usig your code, for σ =, p = 4 : 0 ad M = N = 2 p, compute the approximate solutio give u(x,y) = (2π) 2 + (3π) 2 ex+siy si2πx si 3πy. I leave it to you to compute the appropriate f. Preset a table with the followig data - colum : h; colum 2: u h v h h ; colum 3: u h v h h /h 2 ; colum 4: cpu time; colum 5: (cpu time)/ 2 log, where h x = h y = /. Discuss the treds i each colum. Iclude a copy of your code. ANS: Here is the code for the 2D solver: fuctio [u,x,y] = poiss2d_fft(f,sigma) 2D Poisso eq -(u_xx+u_yy)+ \sigma u = f o [0,]x[0,] Dirichlet BC for u are stored i the first/last-row/colum of f [M N] = size(f); M = M-; N= N-; u = zeros(m+,n+); hx = /M; x = (0:hx:) ; lamx = 2*(-cos(x(2:M)*pi))/(hx^2); hy = /N; y = (0:hy:) ; lamy = 2*(-cos(y(2:N)*pi))/(hy^2); Fourier multiple matrix: lambda_x(i)+lambda_y(j) lamx_p_lamy = (repmat(lamx,,n-) + repmat(lamy,m-,)) ; set RHS ad adjust for possible o-zero Dirichlet BC v = f(2:m,2:n); v(:, ) = v(:, ) + f(2:m, )/(hy^2); v(:,n-) = v(:,n-) + f(2:m,n+)/(hy^2); v(,:) = v(,:) + f(,2:n)/(hx^2); v(m-,:) = v(m-,:) + f(m+,2:n)/(hx^2); solve, put v i u, ad set BC v = dst(dst(dst(dst(v) )./(lamx_p_lamy+sigma)) ); u(2:m,2:n) = v; u(,:) = f(,:); u(m+,:) = f(m+,:); u(:,) = f(:,); u(:,n+) = f(:,n+);

9 Here are the results followed by the discussio ad code: h err_h err_h/h^2 cpu cpu/((^2)log) code: e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-07 sigma = ; p = 4:0; lep = legth(p); h = zeros(lep,); = zeros(lep,); T = zeros(lep,); err_h = zeros(lep,); trials = 5; fac = /((2*pi)^2 + (3*pi)^2); for i = :lep disp(um2str(p(i))) M = 2^p(i); hx = /M; x = 0:hx:; N = 2^p(i); hy = /N; y = 0:hy:; [yy xx]= meshgrid(y,x); u = fac*exp(xx+si(yy)).*si(2*pi*xx).*si(3*pi*yy); f = -fac*exp(xx+si(yy)).*si(3*pi*yy).*... -u_xx ((-4*pi^2)*si(2*pi*xx)+4*pi*cos(2*pi*xx))... -fac*exp(xx+si(yy)).*si(2*pi*xx).*... -u_yy (-9*pi^2*si(3*pi*yy)+6*pi*cos(3*pi*yy).*cos(yy)... -si(3*pi*yy).*si(yy)+si(3*pi*yy).*cos(yy).^2) + u ; f(,:) = u(,:); f(m+,:) = u(m+,:); f(:,) = u(:,); f(:,n+) = u(:,n+); tic; for j=:trials uu = poiss2d_fft(f,sigma); ed stime = toc; h(i) = hx; (i) = M; T(i) = stime/trials; err_h(i) = hx*orm(uu(:)-u(:)); ed format short e disp( ) disp( h err_h err_h/h^2 cpu cpu/((^2)log) ) disp( ) disp( ) disp([h err_h err_h./(h.^2) T T./((.^2).*log2())])