  The central spin model has a wide applicability, it is ideally suited to
describe a small quantum system, for instance a quantum bit, in contact to a
bath of spins, e.g., nuclear spins, or other small quantum systems in general.
According to previous work~[R\"ohrig \textit{et al.}, Phys. Rev. B {\bf 97},
165431 (2018)], a large bath of quantum spins can be described as a bath of
quantum harmonic oscillators. But the resulting quantum model is still far from
being straightforward solvable. Hence we consider a chain representation for
the bosonic degrees of freedom to study how well a truncated Wigner
approximation of the effective model of harmonic oscillators works in
comparison with other approximate and exact methods. Numerically, we examine
the effect of the number of bath spins and of the truncation level, i.e., the
chain length.
