  We investigate the ground state properties of $\mathrm{Na_2IrO_3}$ based on
numerical calculations of the recently proposed {\it ab initio} Hamiltonian
represented by Kitaev and extended Heisenberg interactions. To overcome the
limitation posed by small tractable system sizes in the exact diagonalization
study employed in a previous study (Yamaji $\textit{et al.}$, Phys. Rev. Lett.
$\textbf{113}$, 107201 (2014)), we apply two-dimensional density matrix
renomalization group, and infinite-size tensor-network method. By calculating
at much larger system sizes, we critically test the validity of the exact
diagonalization results. The results consistently indicate that the ground
state of $\mathrm{Na_2IrO_3}$ is a magnetically ordered state with zigzag
configuration in agreement with experimental observations and the previous
diagonalization study. Applications of the two independent methods in addition
to the exact diagonalization study further uncover a consistent and rich phase
diagram near the zigzag phase beyond the accessibility of the exact
diagonalization. For example, in the parameter space away from the $\textit{ab
initio}$ value of $\mathrm{Na_2IrO_3}$ controlled by the trigonal distortion,
we find three phases: (i) an ordered phase with the magnetic moment aligned
mutually in $120$ degrees orientation on every third hexagon, (ii) a
magnetically ordered phase with a $16$-site unit-cell, and (iii) an ordered
phase with presumably incommensurate periodicity of the moment. It suggests
that potentially rich magnetic structures may appear in
$A_\mathrm{2}\mathrm{IrO_3}$ compounds for $A$ other than Na. The present
results also serve to establish the accuracy of the first-principles approach
in reproducing the available experimental results thereby further contribute to
find a route to realize the Kitaev spin liquid.
