# Stability Criterion for this Explicit Scheme

I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as: $$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial u^2}{\partial x} + \frac{\partial uv}{\partial y} = -\frac{\partial p}{\partial x} + \frac{\partial^2 u}{\partial x ^2} + \frac{\partial^2 u}{\partial y ^2}$$

If the time levels in 'physical time, t' goes like n-1, n, n+1, and for that of 'pseudo time, $\tau$ goes like q-1, q, q+1. Then, with the combination of forward and central difference, I can discretize this equation as:

$$\frac{u^{q+1}-u^{q}}{d\tau} + \frac{3u^{q+1}-4u^{n}+u^{n-1}}{2dt} + C = -P + D$$

where C, P and D are the discretized version of convection, pressure and diffusion terms respectively. I use a forward difference for pressure, central difference for diffusion terms.

Iterations are done in pseudo time unless the local steady state is achieved on which I simply assign $u^{n+1} = u^{q+1}$. My concern is of stability at this moment. I seem to infer that both dt and $\tau$ affects the stability but I can not come to a mathematical expression. Therefore I am looking forward to gain some knowledge about how to calculate the CFL number (viscous and advection) for this particular scheme.