Instability of convection

Zeng

I am studying a fluid dynamical problem concerning with instability. In the present stage, the numerical simulation with finite volume method (FVM) is adopted. It seems that the direct numerical simulation is not a very effective method for the study of critical conditions for instability, and the linear stability is very popular. Would you please suggest me on fellowings:

(1) Is there any other effective method based on the origial governing equations except linear stability analysis based on linearized (simplified) governing equations to study the critical condition for instability?

(2) Would you please suggest me some good books and papers to learn linear stability analysis as a starting point.

(3) The basic idea of linear stability analysis seems (maybe I am wrong) that the x is decomposed as basic state x0 and distubance state x' x=x0+x', and x' include a serials of distubance. The flow state after instability should coresspond to the disturbace with largest growth rate. My question is whether this idea can be applied directly to the direct numerical simulation based on original govering equations rather than linearized equations. In other word, a small disturbance including a serials model is imposed in the direct simulation. If this is possible, how to decide effectively which disturbance model is most danger?

Thanks in advance.