Stabilization scheme

Some of the terms in the EDF introduce to the time-dependent propagation of the high-momenta components. In particular, gradient terms U_q^{\Delta\rho} have been identified as a source of such high-k modes. These modes can amplify during the time-dependent propagation and destabilize the integration scheme. To avoid this, we introduced the filtering scheme.

1. compute U_q^{\Delta\rho}(\vec{r}),
2. go to Fourier space U_q^{\Delta\rho}(\vec{k}),
3. apply filter function \tilde{U}_q^{\Delta\rho}(\vec{k})=U_q^{\Delta\rho}(\vec{k})\cdot FD(\frac{k^2}{2m},\mu, T),
4. go back to coordinate space \tilde{U}_q^{\Delta\rho}(\vec{r}) and use it during the time-propagation.

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

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

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 output of the script.

Controlling the filter

The filter can be controlled via input file:

# ------------- HIGH K-WAVES FILTER ---------------
# See: W-BSK Wiki -> Stabilization of the time-dependent code
hkf_mu                  0.9        # mu parameter of the Fermi-Dirac (filtering) function, in Ec units, default=9.99 (disabled) 
hkf_T                   0.01       # T  parameter of the Fermi-Dirac (filtering) function, in Ec units, default=0.01

Benchmark

Below we demonstrate energy conservation quality for the evolution of a nuclei Z=40 immersed in superfluid see of neutron (background density n=0.0086\,\text{fm}^{-3}) for various filters.