How to make wind forecast like Predictwind.com? Which mathematical model has been used?
It seems like that particular website is a commercial forecasting website and requires login/registration information, so I will take your word for the output.

The general equation of motion is described by the Navier-Stokes Equations, which is $$\frac{\partial \vec{u}}{\partial t}+(\vec{u} \cdot\nabla)\vec{u}=-\frac{1}{\rho}\nabla P+g\hat{k}+\nu \nabla^2\vec{u}+f\hat{k}\times\vec{u}$$ where $\vec{u}$ is the atmospheric motion vector (wind + vertical motion), $\rho$ is density, $P$ is pressure, $g$ is gravity, $f$ is the coriolis parameter, and $\nu$ is the kinematic viscosity. 

This very complicated equation currently has no analytic solution (if you have one, you can win $1,000,000 from the Clay Mathematics Institute), but is instead discretized and solved using even heavier mathematics. Additionally, some mathematical tricks are played on the equations to 'simplify' them, but it usually results in other problems.

While 8 km seems like a fine scale, it actually isn't. Smaller phenomenon (called eddies) affect the weather significantly, so research that tries to mimic the effect of those eddies is applied, but there are still problems. Cutting edge research tries to reduce that 8 km (called resolution- think of the resolution of your television) down to tens of meters, but at a significant cost- one that limits the domain.

Likely, the data that your website comes from a weather model, which attempts to solve the above equation. Such models could include the ECMWF, GFS, and other global models.