### Added basic theory (equations essentially) behind PUSPH here

parent 5779ee54
 ... ... @@ -377,7 +377,70 @@ itself is \emph{not} updated. \subsection{Pressure-Energy SPH} \label{sec:sph:pu} Section 2.2.2 of \cite{Hopkins2013}.\\ \tbd Section 2.2.2 of \cite{Hopkins2013} describes the equations for Pressure-Energy (P-U) SPH; they are reproduced here with some more details. P-U SPH depends on the calculation of a smoothed pressure, and follows the evolution of the internal energy, as opposed to the entropy. For P-U, the following choice of parameters in the formalism of \S \ref{sec:derivation} provides convenient properties: \begin{align} x_i =& (\gamma - 1) m_i u_i~, \\ \tilde{x}_i =& 1, \label{eq:sph:pu:xichoice} \end{align} leading to the following requirements to ensure correct volume elements: \begin{align} y_i =& \sum_{j} (\gamma - 1) m_j u_j W_{ij} = \bar{P}_i,\\ \tilde{y}_i =& \sum_{j} W_{ij} = \bar{n}_i, \\ \label{eq:sph:pu:yichoice} \end{align} with the resulting variables representing a smoothed pressure and particle number density. This choice of variables leads to the following equation of motion: \begin{align} \frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t} = -\sum_j (\gamma - 1)^2 m_j u_j u_i &\left[\frac{f_{ij}}{\bar{P}_i} \nabla_i W_{ij}(h_i) ~+ \right. \\ &\frac{f_{ji}}{\bar{P}_j} \nabla_i W_{ji}(h_j) ~+ \\ & \left.\nu_{ij}\bar{\nabla_i W_{ij}}\right]~. \label{eq:sph:pu:eom} \end{align} which includes the Monaghan artificial viscosity term and Balsara switch in the final term. The $h$-terms are given as \begin{align} f_{ij} = 1 - & \left[\frac{h_i}{n_d (\gamma - 1) \bar{n}_i \left\{m_j u_j\right\}} \frac{\partial \bar{P}_i}{\partial h_i} \right] \times \\ & \left( 1 + \frac{h_i}{n_d \bar{n}_i} \frac{\partial \bar{n}_i}{\partial h_i} \right)^{-1} \label{eq:sph:pu:fij} \end{align} with $n_d$ the number of dimensions. In practice, the majority of $f_{ij}$ is precomputed in {\tt hydro\_prepare\_force} as only the curly-bracketed term depends on the $j$ particle. This cuts out on the majority of operations, including expensive divisions. In a similar fashion to \MinimalSPH, the internal energy must also be evolved. Following \cite{Hopkins2013}, this is calculated as \begin{align} \frac{\mathrm{d}u_i}{\mathrm{d}t} = \sum_j (\gamma - 1)^2 m_j u_j u_i \frac{f_{ij}}{\bar{P}_i}(\mathbf{v}_i - \mathbf{v}_j) \cdot \nabla_i W_{ij}(h_i)~. \label{eq:sph:pu:dudt} \end{align} \subsubsection{Time integration} Time integration follows exactly the same scheme as \MinimalSPH. \subsubsection{Particle properties prediction} The prediciton of particle properties follows exactly the same scheme as \MinimalSPH. \subsection{Anarchy SPH} Dalla Vecchia (\textit{in prep.}), also described in section 2.2.2 of \cite{Schaller2015}.\\ ... ...
 ... ... @@ -29,7 +29,7 @@ \section{Equation of state} \input{EoS/eos} \section{Derivation of the Equation of Motion} \section{Derivation of the Equation of Motion}\label{sec:derivation} \input{Derivation/sph_derivation.tex} \section{SPH flavours} ... ...
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