My challenge is to solve the following system of equations, which describe gas combustion in porous media:1) Continuity$\varepsilon \frac{\partial \rho_g}{\partial t} +\frac{\partial}{\partial x} \left(\rho_g u_x\right)=0$2) Darcy law (momentum)$u_x=-\frac{k}{\mu} \frac{\partial p}{\partial x}$3) Equation of state, note the variable temperature$\rho_g=\frac{M_Rp}{RT_g(x)}$4) Energy equation for the gas.5) Energy equation for the solid phaseI have succesfully descritised and solved the case where the velocity, pressure and density are assumed constant, i.e the first three equations drop out. But solving the gasodynamical part proved to be a problem.Applying an upwind scheme to 1) (as was suggested here: A good finite difference for the continuity equation)  yeilds a really harsh stability criteria on the timestep, I am forced to have it as low as 1e-6 with a 1e-2 spacial timestep, even when I take the isothermal case, disregarding the combustion for the time being. And I need at least 1e-3 to resolve the energy equations.The first three equations can be also coupled together to form 6) $ \frac{\partial p}{\partial t} +C\frac{\partial^2}{\partial x^2} \left(p^2 \right)=0$ but only in the isothermal case, so that is of little help. I know that people have solved 1)-5) and 6) before, but I couldn't find a description of the schemes they used. I tryed searching articles on compressible flow in porous media specifically, but those all deal with way more complex models (multiphase, deformable solids etc) and use very complicated solving methods.Could someone suggest a good FD scheme for (1)-(3) or say how the stability criteria is formed if one just uses upwinding like I did?
Finite difference scheme for compressible nonisothermal flow in porous media
finite difference;fluid dynamics