How to derive the $\tau$ in the SUPG method for compressible flow?

Question

I want to understand the formulation and derivation of $\tau$ matrix in the SUPG stablization term, and we sometimes get(there are many variation of SUPG stablization)$$ \begin{aligned} & \tau_c=\min \left(\frac{|\boldsymbol{u}| h^e}{2}\right) \\ & \tau_m=\min \left(\frac{\Delta t}{2 \rho}, \frac{h^e}{2 \rho(|\boldsymbol{u}|+c)}, \frac{m^e\left(h^e\right)^2}{4 \mu}\right) \\ & \tau_e=\min \left(\frac{\Delta t}{2 \rho c_v}, \frac{h^e}{2 \rho c_v(|\boldsymbol{u}|+c)}, \frac{m^e\left(h^e\right)^2}{4 \kappa}\right) \end{aligned} $$ after looking for some more papers, it seems that we can get this via error estimate, in a paper about stablized finite element, it only discuss the case of incompressible N-S equation, so how to derive the $\tau$ stablization matrix for compressible flow? could you please recommend some papers that detail the derivation? thank you

Answer 1

There's a great introduction to SUPG in the book by Elman, Silvester, and Wathen. I would start with that.