All the above numbers helps in maintaining the stability of the numerical behavior of the physics being captured.

Courant number: The Courant number is defined as Cr= epsilon Dt/h

Diffusion number: The diffusion number is defined as S=gama Dt/(h*h)

Grid Péclet number: The Péclet number is defined as Pe = Cr/S .

They are used in Advection Diffusion Equation.When the Péclet number is high, the convection term dominates and when the Péclet number is low the diffusion term

dominates. More discussion may be seen in our paper "Redefined cubic B-splines collocation method for solving convection–diffusion equations" in Applied Mathematical Modelling 36 (2012) 5555–5573.

All the numbers determine a stability of numerical scheme. As common, if they reach some critical limit, the numerical solution begins oscillate in space and time with grid period. That oscillatory solution is non-physical and grows rapidly providing the numerical overflow.

I think, the detailes of numerical scheme stability analysis and recommendations about the scheme choosing can be found in any book about the CFD or heat and mass transfer simulation, e.g.:

"Computational Fluid Dynamics" by K.A. Hoffmann;

"Computational techniques for fluid dynamics" by C.A.J. Fletcher;

"Numerical Heat Transfer and Fluid Flow" by S.V. Patankar;

"The Theory of Difference Schemes" or "Computational heat transfer" by A.A. Samarskii;

"Essential Computational Fluid Dynamics" by O. Zikanov.