I added a preconditioner in my CFD code which originally is used to solve compressible NS equations. I choose the preconditioning matrix proposed by Jack Edwards,'Pc'

Pc= I - t1*R

t1 = (1-beta**2)(gamma -1)/a_sound_speed**2

R is another Matrix.

I just multiply the Pc with the residul vector before using AF to update the solution, the d_density/dt is even worse than no preconditioning,but the C_l & C_d's history is improved so much! I don't know why,who can give me some advice?Thanks!

What do you use for the "sound speed" of an incompressible fluid?

hi: I use a local sound speed a_local**2 = gamma*Pressure/Density.

I still have another question "the local Mach number",how can I get it?

In the code, I use : M_local = V_local / a_free_stream. Is it right?

Should i change it to :

M_local = V_local / a_local

The point of my question was to get you to think about what you are doing. The speed of sound in an incompressible material is infinite (as the stiffness of the material increases, the sound speed increases and becomes infinite for an incompressible material with finite density). You divide by the sound speed in your preconditioner, so that term would go to zero. Also Mach number doesn't have much meaning for an incompressible fluid.

That said, one can reasonably approximate the flow of a real compressible fluid by an unreal incompressible FLOW when the Mach number in the compressible flow is sufficiently small (a value less than .3 Mach is commonly accepted) so that there are no large excursions in density such as occur near shocks.

You should reconsider the question of whether the preconditioner you have chosen is appropriate to the mathematical method you are using. The difficulty in answering the questions you are asking provides a clue. The preconditioner was, no doubt, developed based on physical and mathematical reasoning applied to compressible flow. If you cannot find analogous arguments for the incompressible case, then I would suggest looking at alternative preconditioners.