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3.2.8 Entropy production by artificial viscosity

A possibly way to prevent the violation of the entropy condition (172) is to produce entropy with an extra (artifical) diffusion term added to the CIR flux (185) according to

f_{i+\frac{1}{2}}^n \! \left( \mbox...
... -
\epsilon   \left( v_{i+1}^n - v_{i}^n \right)
\enspace .
\end{displaymath} (187)
However, now the parameter \bgroup\color{DEFcolor}$\epsilon$\egroup has to be tuned to be large enough to give the correct weak solution while not smearing it more than necessary.

The viscosity parameter \bgroup\color{DEFcolor}$\epsilon$\egroup could be replaced e.g. by the function

\epsilon'   \left( v_{i+1}^n - v_{i}^n \right)^2
\end{displaymath} (188)
to confine the extra viscosity to regions where the gradient is really large.