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

$$f_{i+\frac{1}{2}}^n = f_{i+\frac{1}{2}}^n \! \left( \mbox... ... - \epsilon \left( v_{i+1}^n - v_{i}^n \right) \enspace .$$ (187)

However, now the parameter $\epsilon$ has to be tuned to be large enough to give the correct weak solution while not smearing it more than necessary.

The viscosity parameter $\epsilon$ could be replaced e.g. by the function

$$\epsilon = \epsilon' \left( v_{i+1}^n - v_{i}^n \right)^2$$ (188)

to confine the extra viscosity to regions where the gradient is really large.