Does anybody have any information on the stability criteria for the one dimensional Navier Stokes equations discretised with a second order central difference scheme without artificial dissipation? I realise that it's a non-linear stability problem, but are there any useful solutions apart from those gained through the model convection-diffusion equation?

Grid Peclet Number = Pe = (pu)/(G/dx) < 2 where: p=density (kg/m^3) u=velocity, assuming uniform on one cell end to the other cell end (m/s) G=diffusion coeff. (kg/m/s) dx=characteristic cell width

For a given physical situation, if one is in violation of that criteria, then there's no choice but to reduce 'dx'. This will increase resource demands, however.

Artificial dissipation is not a concern in general for 2nd order C.D. schemes; the main issue is balancing stability and resource constraints.