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Full conservative equations

The full conservative equations are shown in eq.(33)-(40)

\begin{displaymath}\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\textbf{v})
\end{displaymath} (33)


\begin{displaymath}\frac{\partial\rho\textbf{v}}{\partial t}=-\nabla\cdot\{\rho
...
...rline{I}-\textbf{B}
\textbf{B}-\frac{1}{2}B^2\underline{I}\}
\end{displaymath} (34)


\begin{displaymath}\frac{\partial e}{\partial t}=-\nabla\cdot\{(e+p)\textbf{v}+
\textbf{E}\times\textbf{B}\}
\end{displaymath} (35)


\begin{displaymath}\frac{\partial\textbf{B}}{\partial t}=-\nabla\times
\textbf{E}
\end{displaymath} (36)


\begin{displaymath}\nabla\cdot\textbf{B}=0
\end{displaymath} (37)


\begin{displaymath}\textbf{E}=-\textbf{v}\times\textbf{B}+
\eta\textbf{j}
\end{displaymath} (38)


\begin{displaymath}\textbf{j}=\nabla\times\textbf{B}
\end{displaymath} (39)


\begin{displaymath}p=(\gamma-1)\{e-\frac{1}{2}\rho v^2-\frac{1}{2}B^2\}
\end{displaymath} (40)

This set of equations allows strict numerical conservation of mass, momentum and energy. But practical application of it shows that, in the region of low \(\beta\), numerical difficulties will be met. Sometimes pressure becomes negative because p becomes the difference of large numbers(here we can also see that normalization has to be combined with other techniques to avoid the difference of large numbers, though normalization itself can prevent most such cases).


next up previous contents
Next: Gas dynamic conservative equations Up: Equations Previous: Non-conservative equations