  Here we present a Lindblad master equation that approximates the Redfield
equation, a well known master equation derived from first principles, without
significantly compromising the range of applicability of the Redfield equation.
Instead of full-scale coarse-graining, this approximation only truncates terms
in the Redfield equation that average out over a time-scale typical of the
quantum system. The first step in this approximation is to properly renormalize
the system Hamiltonian, to symmetrize the gains and losses of the state due to
the environmental coupling. In the second step, we swap out an arithmetic mean
of the spectral density with a geometric one, in these gains and losses,
thereby restoring complete positivity. This completely positive approximation,
GAME (geometric-arithmetic master equation), is adaptable between its
time-independent, time-dependent, and Floquet form. In the exactly solvable,
three-level, Jaynes-Cummings model, we find that the error of the approximate
state is almost an order of magnitude lower than that obtained by solving the
coarse-grained stochastic master equation. As a test-bed, we use a
ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling
between up to 25-spins, and study the differences between various master
equations. We find that GAME has the highest accuracy per computational
resource.
