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Daniel J. Price, Guillaume Laibe, Erratum: A fast and explicit algorithm for simulating the dynamics of small dust grains with smoothed particle hydrodynamics, Monthly Notices of the Royal Astronomical Society, Volume 454, Issue 3, 11 December 2015, Page 2320, https://doi.org/10.1093/mnras/stv2125
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errata, addenda, hydrodynamics, methods: numerical, protoplanetary discs, dust, extinction, ISM: kinematics and dynamics
After publication of Price & Laibe (2015), we discovered a sign error in the second term on the right-hand side of equation (4). The equation should read: (1)The sign error was propagated to equations (13), (45), (A1), (A2) and on both sides of (A12). The sign was given correctly in equation (27). The correct sign was also used in the code, so there was no adverse effect on the test results or conservation properties. We further note that the corresponding equation can be found with the correct sign in Laibe & Price (2014).
\begin{equation}
\frac{{\rm d}u}{{\rm d}t} = -\frac{P}{\rho _{\rm g}}(\nabla \cdot \boldsymbol {v}) + \epsilon t_{\rm s} (\Delta \boldsymbol {f}\cdot \nabla u) + \Lambda _{\rm heat} - \Lambda _{\rm cool}.
\end{equation}
Specifically, equation (13) in Price & Laibe (2015) should read: (2)equation (45) should read: equation (A1) should read: (3)equation (A2) should read: (4)and equation (A12) should read : (5)
\begin{equation}
\frac{{\rm d} u}{{\rm d} t} = -\frac{P}{\rho _{\rm g}} (\nabla \cdot {\boldsymbol {v}}) + \frac{\epsilon t_{\rm s}}{\rho _{\rm g}} \left( \nabla P \cdot \nabla u \right) + \Lambda _{\rm heat} - \Lambda _{\rm cool},
\end{equation}
\begin{eqnarray*}
\frac{{\rm d}u_{a}}{{\rm d}t} & = & \frac{1}{\Omega _{a} (1 - \epsilon _{a})\rho _{a}^{2}} \sum _{b} m_{b} (P_{a} + q^{AV}_{ab, a}) \left( \boldsymbol {v}_{a} - \boldsymbol {v}_{b}\right)\cdot \nabla _{a} W_{ab} (h_{a}) \nonumber \\
& - & \frac{1}{2(1-\epsilon _{a})\rho _{a}}\sum _{b} \frac{m_{b}}{\rho _{b}} (u_{a} - u_{b}) (D_{a} + D_{b}) (P_{a} - P_{b}) \frac{\overline{F}_{ab}}{\vert r_{ab}\vert }, \nonumber
\end{eqnarray*}
\begin{equation}
-\frac{1}{2(1 - \epsilon _{a})\rho _{a}}\sum _{b} \frac{m_{b}}{\rho _{b}} (u_{a} - u_{b}) (D_{a} + D_{b}) (P_{a} - P_{b}) \frac{F_{ab}}{\vert r_{ab} \vert },
\end{equation}
\begin{equation}
\frac{\epsilon t_{\rm s}}{\rho _{\rm g}} \nabla P \cdot \nabla u,
\end{equation}
\begin{equation}
\frac{D}{\rho _{\rm g}} (\nabla P \cdot \nabla u) = \frac{\epsilon t_{\rm s}}{\rho _{\rm g}} \nabla P \cdot \nabla u.
\end{equation}
REFERENCES
© 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
Issue Section:
Erratum
