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--- gibson:juliablog:kuramoto_sivashinksy [2017/07/09 06:22]
gibson [The results]
+++ gibson:juliablog:kuramoto_sivashinksy [2017/07/09 07:30] (current)
gibson
@@ Line 17: / Line 17: @@
 \end{eqnarray*}
-on a 1d periodic domain $[0, L_x]$, with $x$ space and $t$ time, and where subscripts indicate differentiation. The algorithm uses a Fourier decomposition in space and 2nd-order Crank-Nicolson, Adams-Bashforth semi-implicit finite-differencing in time. I implemented the same algorithm in Python, Matlab, C++, and in two forms in Julia. A detailed description of the algorithm is given below.
+on a 1d periodic domain $[0, L_x]$, with $x$ space and $t$ time, and where subscripts indicate differentiation. The algorithm uses a Fourier decomposition in space and 2nd-order Crank-Nicolson, Adams-Bashforth semi-implicit finite-differencing in time, with collocation computation of the nonlinear term $u u_x$. I implemented the same algorithm in Python, Matlab, C++, and in two forms in Julia. The codes and a detailed description of the algorithm is given below.
 ==== The results ====
-First, the results. The left plot shows execution time of 3200 time steps of the algorithm as a function of $N_x$, the number of gridpoints in the Fourier decomposition. The dominant cost of the algorithm should be the FFTs
+**The left plot** shows execution time of 3200 time steps of the algorithm as a function of $N_x$, the number of gridpoints in the Fourier decomposition. The dominant cost of the algorithm should be the FFTs, which should scale as $N_x \log N_x$. All the codes use the same FFTW libraries, so ideally, they should all collapse onto the same $N_x \log N_x$ line as linear and fixed-size overheads costs decrease relative to that.
 {{:gibson:juliablog:cputime.png?400|}} {{:gibson:juliablog:timeloc.png?400|}}
 The right plot shows

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