  We study the square-lattice extended Hubbard model with on-site $U$ and
nearest-neighbor $V$ interactions by exact diagonalization. We show that
non-equilibrium quench dynamics can help determine the equilibrium phase
transition boundaries, which agree with the calculations of fidelity metric,
dynamical structure factor, and correlation function. At half filling, the
phase diagrams in the strong-coupling regime include spin density wave and
$d_{x^2-y^2}$-wave superconductivity at large positive $U$, charge density wave
(extended $s^*$-wave superconductivity) at large positive (negative) $V$, and
$s$-wave superconductivity at large negative $U$ with vanishing $V$. The
energies of different particle sectors also help determine the phase separation
region. With carrier doping, charge fluctuation result in strong competition
between different orders, making it more difficult to identify the leading
instability on finite-size cluster. Nevertheless, the more exotic $p$-wave
superconducting pairing is found to be enhanced when the system is heavily
overdoped by $37.5\%-50\%$ holes, especially in interaction parameter range
relevant to the cuprate superconductors.
