connection between the step-sizes and numerical-speed
 

Is there any connection between the step-size and the wave speed ?
for example, we have hyperbolic conservation equation u_t+f(u)_x=0, If we write it's numerical scheme for non-uniform (2p+1) points then how we will write the expression for f'(u) for non-uniform mesh ? I would also like to know of some references regarding this.

Not clear to me why do you define "numerical" a term that appears in the PDE ... Could you detail better you question? Is that focused on hyperbolic equations?

Yes Sir, the question is based on the hyperbolic equation. In the case of uniform mesh the finite difference scheme under consideration are of (2p+1) points in conservative form,
v_j^{n+1}=H(v_{j-p}^n, ... , v_{j+p}^n):=v_j^n-k/h [F(v_{j-p+1}, ... , v_{j+p}^n) - F(v_{j-p}^n, ... ,v_{j+p-1})].
The expression for a(u) is mentioned in Page No. (303) of the given Reference.

Reference: A. Harten, J. M. Hyman and P. D. Lax: On Finite Difference Approximation and Entropy Conditions for shocks


How we will write the expression for wave- speed a(u) for non-uniform mesh-points ?

One of the best textbook I can address is Leveque, Finite Volume for Hyperbolic Problems.

In a conservative method you do not write the equation in a quasi linear form, that is you evaluate the flux f(u) not the term df/du*du/dx