Commit c7960649 authored by Josh Borrow's avatar Josh Borrow
Browse files

Added basic theory (equations essentially) behind PUSPH here

parent 5779ee54
...@@ -377,7 +377,70 @@ itself is \emph{not} updated. ...@@ -377,7 +377,70 @@ itself is \emph{not} updated.
\subsection{Pressure-Energy SPH} \subsection{Pressure-Energy SPH}
\label{sec:sph:pu} \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} \subsection{Anarchy SPH}
Dalla Vecchia (\textit{in prep.}), also described in section 2.2.2 of Dalla Vecchia (\textit{in prep.}), also described in section 2.2.2 of
\cite{Schaller2015}.\\ \cite{Schaller2015}.\\
......
...@@ -29,7 +29,7 @@ ...@@ -29,7 +29,7 @@
\section{Equation of state} \section{Equation of state}
\input{EoS/eos} \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} \input{Derivation/sph_derivation.tex}
\section{SPH flavours} \section{SPH flavours}
......
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