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Non-conservative equations

Eq.(26)-(32) are non-conservative equations


\begin{displaymath}\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\textbf{v})
\end{displaymath} (26)


\begin{displaymath}\frac{\partial\textbf{v}}{\partial t}=-(\textbf{v}\cdot
\nab...
...c{1}{\rho}\nabla p+\frac{1}{\rho}
\textbf{j}\times\textbf{B}
\end{displaymath} (27)


\begin{displaymath}\frac{\partial p}{\partial t}=-(\textbf{v}\cdot\nabla)p-
\gamma p\nabla\cdot\textbf{v}
\end{displaymath} (28)


\begin{displaymath}\frac{\partial\textbf{B}}{\partial t}=-\nabla\times
\textbf{E}
\end{displaymath} (29)


\begin{displaymath}\nabla\cdot\textbf{B}=0
\end{displaymath} (30)


\begin{displaymath}\textbf{E}=-\textbf{v}\times\textbf{B}+
\eta\textbf{j}
\end{displaymath} (31)


\begin{displaymath}\textbf{j}=\nabla\times\textbf{B}
\end{displaymath} (32)

In a real physical system, we should have the conservation of momentum, energy and mass. But this set of equations can not guarantee strict numerical conservation of momentum and energy, though it can assure the conservation of mass. At the same time, it also has numerical difficulties with convective derivatives. The practical application of this set of equations also shows that non-conservative equations lead to numerical difficulties with strong shocks and the errors in the Rankine-Hugoniot conditions and shock speed. So it is improper for us to use it in shock related problems.


next up previous contents
Next: Full conservative equations Up: Equations Previous: Equations