To examine the numerical diffusion you need to do the standard consistency analysis in the space variable, that is, you insert the exact solution into your numerical scheme, expand terms of the form $f(x_i+h)$ into a Taylor series, and estimate the terms that you cut off in your approximation. The terms that are of the form $c_d \frac{\partial ^2 f}{\partial x^2}$ and that do not cancel out, are then responsible for numerical diffusion.
This is straight forward for discretizations of convection terms, but I am not sure what this will give for your square-root example.
Numerical dispersion is harder to analyse. It occurs when your discretization 'supports' so-called checkerboard modes, i.e. unwanted decoupling of the variables that become visible as oscillations in the numerical solution.
Both issues are well understood for upwind discretization (leading to numerical diffusion) and discretization via central differences (leading to numerical dispersion) in the Finite Differences or Finite Volumes approximation of convection-diffusion equations. Maybe, you can transfer some ideas from there to your particular case.