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====== Differences ====== This shows you the differences between two versions of the page.

--- gibson:teaching:spring-2014:iam950:hw1 [2014/02/27 08:54]
gibson
+++ gibson:teaching:spring-2014:iam950:hw1 [2014/02/27 09:39] (current)
gibson
@@ Line 111: / Line 111: @@
 in that the 8 represents the sum of the k=4 and k=-4 cosine modes, with no representation
 of the k=4 and k=-4 sin modes. Without going into too much detail on this, Fourier spectral
-PDE codes typically just zero the highest-order mode at every time step.
+PDE codes typically just zero the highest-order mode at every time step. We can ignore this
+subtlety.
+Based on the above, here's a short example of how to do Fourier differentiation in Matlab on
+a [0,L] periodic domain. This should be plenty to get you started for the time-stepping codes.
+<code>
+% A demonstration of Fourier differentiation in Matlab
+% Parameters
+L = 10;
+N = 16;
+x = L/N*(0:N-1);
+% Set up u(x) and du/dx for some known L-periodic function
+a = 2*pi/L
+u    =   sin(a*x) + 0.2*cos(a*2*x);
+dudx = a*cos(a*x) - 0.4*a*sin(a*2*x);
+% Now compute dudx via FFT
+k = [0:N/2-1 -N/2:-1];
+alpha = 2*pi*k/L;
+D = i*alpha;
+dudx_fft = real(ifft(D.*fft(u)));
+% Plot everybody
+plot(x,u, 'b', x,dudx, 'g', x, dudx_fft, 'r.')
+legend('u','du/dx','du/dx via FFT','location','southwest')
+xlabel('x')
+</code>