### Complete the discussion of the truncated MAC estimator

parent bfa215a1
 ... ... @@ -189,7 +189,25 @@ f_{\rm MAC}(r) = \right. \label{eq:fmm:f_mac} \end{align} This esimator is shown as a dot-dashed line on Fig. \ref{fig:fmm:mac_potential} and obeys the relation $f_{\rm SWIFT}(r) \leq f_{\rm MAC}(r) \leq 1/r^2$, with $f_{\rm SWIFT}(r)$ the true truncated and softened forces (green line). Since it is made of constants and even powers of the distance, computin this term is much cheaper than the true forces. This esimator is shown as a dot-dashed line on Fig. \ref{fig:fmm:mac_potential} and obeys the relation $f_{\rm SWIFT}(r) \leq f_{\rm MAC}(r) \leq 1/r^2$, with $f_{\rm SWIFT}(r)$ being the true truncated and softened norm of the gravity forces the code solves for (green line). We use this expression in the multipole acceptance criterion instead of the $1/|\mathbf{R}|$ term: \begin{equation} \tilde{E}_{BA,p} M_Bf_{\rm MAC}(|\mathbf{R}|) < \epsilon_{\rm FMM} \min_{a\in A}\left(|\mathbf{a}_a|\right). \label{eq:fmm:mac_f_mac} \end{equation} The same change is applied to the MAC used of the M2P kernel (eq. \ref{eq:fmm:mac_m2p}). In the non-truncated un-softened case, this expression reduces to \citep{Dehnen2014} one. Using this expression instead of the simpler Newtonian one only makes a difference in simulations where a lot of particles cluster below the scale of the softening, which is often the case for hydrodynamical simulations including radiative cooling processes. The use of this term over the simpler $1/r^2$ estimator is a runtime parameter.
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