  The reduced dynamics of an open quantum system obtained from an underlying
microscopic Hamiltonian can in general only approximately be described by a
time local master equation. The quality of that approximation depends primarily
on the coupling strength and the structure of the environment. Various such
master equations have been proposed with different aims. Choosing the most
suitable one for a specific system is not straight forward. By focusing on the
accuracy of the reduced dynamics we provide a thorough assessment for a
selection of methods (Redfield Equation, Quantum Optical Master Equation,
Coarse-Grained Master Equation, a related dynamical map approach and a
partial-secular approximation). Whether or not an approach guarantees
positivity we consider secondary, here. We use two qubits coupled to a
Lorentzian environment in a spin-boson like fashion modeling a generic
situation with various system and bath time scales. We see that, independent of
the initial state, the simple Redfield Equation with time dependent
coefficients is significantly more accurate than all other methods under
consideration. We emphasize that positivity violation in the Redfield formalism
becomes relevant only in a regime where any of the perturbative master
equations considered here are rendered invalid anyway. This implies that the
loss of positivity should in fact be welcomed as an important feature: it
indicates the breakdown of the weak coupling assumption. In addition we present
the various approaches in a self-contained way and use the behavior of their
errors to provide further insight into the range of validity of each method.
