Fast Fourier Transform

We solve the Poisson equation subject to periodic boundary conditions. The PPM FFT-based Poisson solver performs the following steps:

  • Transformation of the right hand side of the Poisson Equation to Fourier space.
  • Solving the Poisson Equation in Fourier space.
  • Transformation of the Fourier space solution to real space.

The data array to be transformed is optimally stored if the transformation operates along the leading dimension of the array (unit stride). Therefore, the data is transposed if necessary before performing the individual one-dimensional FFTs by using ppm pencil topologies and field mapping routines . The complete three-dimensional FFT thus consists of the following steps:

  1. If necessary, map the field data onto a temporary x-pencil topology. The field topology on input is detected automatically.
  2. Perform a one-dimensional FFT in x-direction.
  3. Map the field data onto a temporary y-pencil topology and transpose the data.
  4. Perform a one-dimensional FFT in y-direction.
  5. Map the field data onto a temporary z-pencil topology and transpose the data.
  6. Perform a one-dimensional FFT in z-direction.

The inverse transformation involves the reverse process.

 

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