Fast Fourier Transforms

This program aims to solve the diffusione equation

$$ \frac{\partial c(\mathbf{r}, t)}{\partial t}=\nabla\cdot[D(\mathbf{r})\nabla c(\mathbf{r}, t)] $$

where $c(\mathbf{r}, t)$ concentration in $\mathbb{R}^3\times \mathbb{R}$ and $D(\mathbf{r})$ diffusion coefficent in $\mathbb{R}^3$, using the Fourier transform of it.

Recall that given a function $f(x)\in L^1(\mathbb{C})$ it Fourier transform $F(q)$ can be defined as:

$$ F(q)=\int_{-\infty}^{+\infty}e^{iqx}f(x)dx $$

and it inverse as:

$$ f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{iqx}F(q)dq. $$

Therefore, taking the transform of the $f(x)$ differential w. r. t. $x$ yields:

$$ \begin{aligned} \frac{d}{dx}f(x)&=\frac{1}{2\pi}\frac{d}{dx}\int_{-\infty}^{+\infty}e^{iqx}F(q)dq \\ &=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{iqx}(iq)F(q)dq. \end{aligned} $$

i. e., the differential operator

$$ \frac{d}{dx}[\cdot] $$

in the direct space can be associated to the $iq[\cdot]$ operator in the reciprocal space. We make use of this property.

Dependencies

• mpi compliant library
• FFTW library
• gnuplot

Compilation

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Use:

make [mode]

with [mode]:

• left blank to compile an homemade version (that is, a version which exploits only locally the FFTW functions and handles explicitly the communication between processes) to execute a FFT on a 3D grid
• fftw3_mpi to compile the FFTW routine which runs in parallel on a distributed 3D grid

This will produce the ./[version]diffusion.x executable

Execution

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Use:

make [version]run [iters=%d] [nx=%d] [ny=%d] [nz=%d] [dt=%d] [prc=%d]

where:

• nx, ny and nz dimensions of the grid
• iters number of iterations
• dt dimension of the single iteration time step
• prc number of processes

Test

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To test it is possible to pass the debug=yes flag while making

make [version] debug=yes

and then run the [version]multiplication_debug.x executable using mpirun. It is also supported the command:

make [version]run debug=yes

which will compile (if necessary) and run immediately after.

Plot

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Use make plot to plot pretty things