Stabilization scheme

Some terms in the EDF introduce time-dependent propagation of the high-momentum components. Examples are terms that involve division by density and may generate noise 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 filter 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. See here for 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