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Gas dynamic conservative equations

Eq.(41)-(48) are gas dynamic conservative equations.


\begin{displaymath}\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\textbf{v})
\end{displaymath} (41)


\begin{displaymath}\frac{\partial\rho\textbf{v}}{\partial t}=-\nabla\cdot(\rho\textbf{v}
\textbf{v}+p\underline{I})+\textbf{j}\times\textbf{B}
\end{displaymath} (42)


\begin{displaymath}\frac{\partial e}{\partial t}=-\nabla\cdot\{(e+p)\textbf{v}\}+
\textbf{j}\cdot\textbf{E}
\end{displaymath} (43)


\begin{displaymath}\frac{\partial\textbf{B}}{\partial t}=-\nabla\times
\textbf{E}
\end{displaymath} (44)


\begin{displaymath}\nabla\cdot\textbf{B}=0
\end{displaymath} (45)


\begin{displaymath}\textbf{E}=-\textbf{v}\times\textbf{B}+
\eta\textbf{j}
\end{displaymath} (46)


\begin{displaymath}\textbf{j}=\nabla\times\textbf{B}
\end{displaymath} (47)


\begin{displaymath}p=(\gamma-1)\{e-\frac{1}{2}\rho v^2\}
\end{displaymath} (48)

This set of equations is a compromise between the other two sets of equations. It allows strict numerical conservation of mass, momentum and plasma energy, but no strict conservation of total energy. In low \(\beta\) regions, it poses no difficulty in simulation. At the same time, the gas dynamic conservative equations can be combined with the full conservative scheme by integrating both energy equations and using a \(\beta\) 'switch'.


next up previous contents
Next: Time and space discretization Up: Equations Previous: Full conservative equations