  Quantum master equations are commonly used to model the dynamics of open
quantum systems, but their accuracy is rarely compared with the analytical
solution of exactly solvable models. In this work, we perform such a comparison
for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an
analytical solution is available in the one-excitation subspace. We consider
the non-Markovian time-convolutionless master equation up to the second
(Redfield) and fourth orders as well as three types of Markovian master
equations: the coarse-grained, cumulant, and standard rotating-wave
approximation (RWA) Lindblad equations. We compare the exact solution to these
master equations for three different spectral densities: impulse, Ohmic, and
triangular. We demonstrate that the coarse-grained master equation outperforms
the standard RWA-based Lindblad master equation for weak coupling or high qubit
frequency (relative to the spectral density high-frequency cutoff $\omega_c$),
where the Markovian approximation is valid. In the presence of non-Markovian
effects characterized by oscillatory, non-decaying behavior, the TCL
approximation closely matches the exact solution for short evolution times (in
units of $\omega_c^{-1}$) even outside the regime of validity of the Markovian
approximations. For long evolution times, all master equations perform poorly,
as quantified in terms of the trace-norm distance from the exact solution. The
fourth-order time-convolutionless master equation achieves the top performance
in all cases. Our results highlight the need for reliable approximation methods
to describe open-system quantum dynamics beyond the short-time limit.
