In the equation, v is the velocity, and Φ is the gravitational potential given by Poisson's Equation. These two coupled equations are solved in an expanding background Universe, which is governed by the Friedmann equations, after determining the initial conditions of dark matter particles. The conventional method employed for initializing positions and velocities of dark matter particles involves moving particles within a uniform Cartesian lattice or a glass-like particle configuration. This is done by using a linear theory approximation or a low-order perturbation theory.