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# Stabilization scheme
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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.
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1. compute $`U_q^{\Delta\rho}(\vec{r})`$,
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2. go to Fourier space $`U_q^{\Delta\rho}(\vec{k})`$,
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3. apply filter function $`\tilde{U}_q^{\Delta\rho}(\vec{k})=U_q^{\Delta\rho}(\vec{k})\cdot FD(\frac{k^2}{2m},\mu, T)`$,
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4. go back to coordinate space $`\tilde{U}_q^{\Delta\rho}(\vec{r})`$ and use it during the time-propagation.
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As the filter function, we use Fermi-Dirac function:
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```math
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FD(e_k,\mu, T)=\frac{1}{\exp[\frac{e_k-\mu}{T}]+1}
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```
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# Testing script
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You can use the attached script [tools/high-frequency-filter.py](https://gitlab.fizyka.pw.edu.pl/wtools/w-bsk/-/blob/devel/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.
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# Controlling the filter
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The filter can be controlled via input file:
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```bash
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# ------------- HIGH K-WAVES FILTER ---------------
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# See: W-BSK Wiki -> Stabilization of the time-dependent code
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hkf_mu 0.9 # mu parameter of the Fermi-Dirac (filtering) function, in Ec units, default=9.99 (disabled)
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hkf_T 0.01 # T parameter of the Fermi-Dirac (filtering) function, in Ec units, default=0.01
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```
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# Benchmark
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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.
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\ No newline at end of file |
