Mode_Correction.md

Correction of the modes

Both versions of the solver (cell-centred or node-centred) use different FFT types to impose the correct boundary conditions. In some cases, points in memory (in general, the first or the last) are not taken into account by fftw as they are null, however they are required when performing the spectral convolution with the Green's function. Therefore, we postprocess the data after the 1D FFTs to ensure that the Green's function and the RHS are evaluated at the same wavenumber. The following tables detail those corrections.

Here is the list of all the post-processings:

• NULL_FIRST_POINT resets the first point in memory to 0
• NULL_FIRST_POINT resets the last point in memory to 0
• ENFORCE_PERIOD corrects the last point and enforce the periodicity of the domain

All of this variables are set in the FFTW_plan_dim class (either node or cell).

Cell-centred case

In the cell-centred case, the RHS is computed in a cell-centred framework while the Green's functions are computed in a node-centred framework as it characterizes the influence of a source located in x = 0 on any other location in the domain. This is the main reasons to apply corrections.

BCs	Direction	RHS FFT types	RHS Correction	Green's function Correction
PER - PER	Forward	DFT	POSTPRO_NONE	POSTPRO_NONE
	Backward	DFT	POSTPRO_NONE	POSTPRO_NONE
ODD - ODD	Forward	DST-II - RODFT10	NULL_FIRST_POINT	POSTPRO_NONE
	Backward	DST-III - RODFT01	NULL_LAST_POINT	POSTPRO_NONE
ODD - EVEN	Forward	DST-IV - RODFT11	POSTPRO_NONE	modes shifted by 1/2
	Backward	DST-IV - RODFT11	POSTPRO_NONE	modes shifted by 1/2
EVEN - ODD	Forward	DCT-IV - REDFT11	POSTPRO_NONE	modes shifted by 1/2
	Backward	DCT-IV - REDFT11	POSTPRO_NONE	modes shifted by 1/2
EVEN - EVEN	Forward	DCT-II - REDFT10	NULL_LAST_POINT	POSTPRO_NONE
	Backward	DCT-III - REDFT01	POSTPRO_NONE	POSTPRO_NONE

Node-centred case

In the node-centred case, both the RHS and the Green's function are computed in a node-centred framework. However, FFTW discard some points hen dealing with even boundary, as they do not carry any information (the boundary is null). We therefore have to correct the output field of FFTW to have the RHS and GF evaluated at the same wavenumber.

BCs	Direction	RHS FFT types	RHS Correction	Green's function Correction
PER - PER	Forward	DFT	POSTPRO_NONE	POSTPRO_NONE
	Backward	DFT	POSTPRO_NONE	POSTPRO_NONE
ODD - ODD	Forward	DST-I - RODFT00	NULL_FIRST_POINT + NULL_LAST_POINT	POSTPRO_NONE
	Backward	DST-I - RODFT00	NULL_FIRST_POINT + NULL_LAST_POINT	POSTPRO_NONE
ODD - EVEN	Forward	DST-III - RODFT01	NULL_LAST_POINT	modes shifted by 1/2
	Backward	DST-II - RODFT10	NULL_FIRST_POINT	modes shifted by 1/2
EVEN - ODD	Forward	DCT-IV - REDFT11	NULL_LAST_POINT	modes shifted by 1/2
	Backward	DCT-IV - REDFT11	NULL_LAST_POINT	modes shifted by 1/2
EVEN - EVEN	Forward	DCT-I - REDFT00	POSTPRO_NONE	POSTPRO_NONE
	Backward	DCT-I - REDFT00	POSTPRO_NONE	POSTPRO_NONE