Benchmarking a simple PDE solver in Julia and other languages

Julia

Julia is an innovative new programming language that promises to revolutionize scientific computing. In a nutshell, it is

High-level, dynamic, and general-purpose, like Python
As fast as compiled C, roughly
Aimed squarely at numerics, with libraries and ease-of-use comparable to Matlab

Julia's main innovation is a carefully-designed type system combined with just-in-time compilation. The combination allows high-level user code to be compiled to machine-code on-the-fly.

The benchmark algorithm

The benchmark algorithm here is a simple time-integration of the Kuramoto-Sivashniksy equation

$\begin{eqnarray*} u_t = -u_{xx} - u_{xxxx} - u u_x \end{eqnarray*}$

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, 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

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.