  Employing the publicly available CosmoLattice code, we conduct numerical
simulations of a domain wall network and the resulting gravitational waves
(GWs) in a radiation-dominated Universe in the $Z_2$-symmetric scalar field
model. In particular, the domain wall evolution is investigated in detail both
before and after reaching the scaling regime, using the combination of
numerical and theoretical methods. We demonstrate that the total area of closed
walls is negligible compared to that of a single long wall stretching
throughout the simulation box. Therefore, the closed walls are unlikely to have
a significant impact on the overall network evolution. This is in contrast with
the case of cosmic strings, where formation of loops is crucial for maintaining
the system in the scaling regime. To obtain the GW spectrum, we develop a
technique that separates physical effects from numerical artefacts arising due
to finite box size and non-zero lattice spacing. Our results on the GW spectrum
agree well with Refs. [29, 30], which use different codes. Notably, we observe
a peak at the Hubble scale, an exponential falloff at scales shorter than the
wall width, and a plateau/bump at intermediate scales. We also study
sensitivity of obtained results on the choice of initial conditions. We find
that different types of initial conditions lead to qualitatively similar domain
wall evolution in the scaling regime, but with important variations translating
into different intensities of GWs.
