Commit 09d36187 by Josh Borrow

### Minor fixes to the formatting of equations in the theory for P-U

parent c7960649
 ... ... @@ -386,14 +386,14 @@ 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, 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, \\ \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 ... ... @@ -401,8 +401,8 @@ 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[\frac{f_{ij}}{\bar{P}_i} \nabla_i W_{ij}(h_i) ~+ \right. \nonumber \\ &\frac{f_{ji}}{\bar{P}_j} \nabla_i W_{ji}(h_j) ~+ \nonumber \\ & \left.\nu_{ij}\bar{\nabla_i W_{ij}}\right]~. \label{eq:sph:pu:eom} \end{align} ... ... @@ -411,9 +411,10 @@ 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} 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] \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 ... ...
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