diff --git a/theory/Multipoles/fmm_summary.tex b/theory/Multipoles/fmm_summary.tex index ae92a07bd0311a522c8b64a5d19d9907869ba548..ba3659682b589c275c8394ccf5d8b6016402fd38 100644 --- a/theory/Multipoles/fmm_summary.tex +++ b/theory/Multipoles/fmm_summary.tex @@ -315,8 +315,9 @@ the field tensors in cell $B$ but the multipole in $B$ cannot be used to compute the $\mathsf{F}$ values of cell $A$ and vice versa. This affects the tree walk by breaking the symmetry and potentially leading to cells of different sizes interacting. \\ -In the specific case of the M2P kernel, we have $\rho_A = 0$, which -simplifies some of the expressions above. In this case, at order $p$, we get: +For the M2P kernel, the sink is a single particle $a$ and hence +$\rho_A = 0$, which simplifies some of the expressions above. In this +case, at order $p$, we get: \begin{equation} E_{BA,p} = \frac{P_{B,p}}{M_B |\mathbf{R}|^p}, \qquad \tilde{E}_{BA,p} = 8E_{BA,p} \nonumber @@ -325,9 +326,8 @@ Note that, in this case, only the power term of the order of the scheme appears; not a sum over the lower-order ones. This leads to the following MAC for the M2P kernel: \begin{equation} - 8\frac{P_{B,p}}{|\mathbf{R}|^{p+2}} < \epsilon \min_{a\in - A}\left(|\mathbf{a}_a|\right) \quad \rm{and} \quad \frac{\rho_B} - {|\mathbf{R}|} < 1. + 8\frac{P_{B,p}}{|\mathbf{R}|^{p+2}} < \epsilon |\mathbf{a}_a| \quad + \rm{and} \quad \frac{\rho_B} {|\mathbf{R}|} < 1. \label{eq:fmm:mac_m2p} \end{equation} The value of $\epsilon$ could in principle be different than the one @@ -337,7 +337,7 @@ $2$ and the approximation $P_{B,p} \approx M_B \rho_B^p$, we get \begin{equation} 8\frac{M_B}{|\mathbf{R}|^2}\left(\frac{\rho_B}{|\mathbf{R}|}\right)^2 - < \epsilon \min_{a\in A}\left(|\mathbf{a}_a|\right) \nonumber + < \epsilon |\mathbf{a}_a| \nonumber \end{equation} for our MAC. This is the same expression as the adaptive opening angle used by \gadget \cite[see eq.18 of][]{Springel2005} up to