4.1.3 Positivity of pressure

The pressure $P\left( e_{\rm i}, \rho \right)$ depends on the conserved quantities via

$$P\left( \rho , \enspace \rho v_{} , \enspace\rho e_{\rm ik}... ..._{} \right)^2}{2 \rho^2} , \enspace \rho \right) \enspace .$$ (198)

An overshoot in velocity may lead to negative pressure (or an attempt to access values beyond the limits of a tabulated equation of state).

• This restriction is so severe that in some cases the conservation of total energy might be given up in favor of a (non-conservative) formulation where the positivity of the internal energy is guaranteed.
• The total variation of e.g. density, pressure,...of the true solution can increase $\rightarrow$ $TV$ stability arguments no longer hold.