Stabilization scheme

Some of the terms in the EDF introduce the time-dependent propagation of the high-momenta components. Examples are terms that contain division by density and may lead to noise generation in regions where density vanishes. These modes can amplify during the time-dependent propagation and destabilize the integration scheme. To avoid this, we introduced the filtering scheme.

1. compute mean-field V_\sigma(\vec{r}),
2. go to Fourier space V_\sigma(\vec{k}),
3. apply filter function \tilde{V}_\sigma(\vec{k})=V_\sigma(\vec{k})\cdot FD(\frac{k^2}{2m},\mu, T),
4. go back to coordinate space \tilde{V}_\sigma(\vec{r}) and use it during the time-propagation.

As the filter function, we use the Fermi-Dirac function:

FD(e_k,\mu, T)=\frac{1}{\exp[\frac{e_k-\mu}{T}]+1}

The same procedure can be used to folder noise that is generated in the pairing potential \Delta(\vec{r}).

Testing script

You can use the attached script tools/high-frequency-filter.py to test the impact of the filtering scheme on the input signal. Below is an example of the script output.

Controlling the filter

The filter can be controlled via the input file:

# -------------- HIGH K-WAVES FILTER ----------------
# See: Wiki -> Stabilization of the time-dependent code
# hkf_mode                1          # 0 - no noise filtering (default)
                                     # 1 - noise filtering for mean-fields only
                                     # 2 - noise filtering of mean-fields and pairing field
# hkf_mu                  0.9        # mu parameter of the Fermi-Dirac (filtering) function, in Ec units, default=0.9
# hkf_T                   0.02       # T  parameter of the Fermi-Dirac (filtering) function, in Ec units, default=0.02