We present in this work a mathematical and numerical analysis of a master equation modeling the interaction of a system with a noisy environment in the particular context of open quantum systems. We show that a transformed master equation can have a simpler structure than a Wigner model by expressing it in convenient coordinates, having a reduced computational cost in comparison to the computation of a Wigner-Fokker-Planck model of the same system. We then present specifics of Discontinuous Galerkin (DG) numerical schemes, such as Nonsymmetric Interior Penalty Galerkin - Discontinuous Galerkin (NIPG-DG) and Local Discontinuous Galerkin (LDG), adequate for the convection-diffusion system obtained, which will let us solve computationally this transformed system. We present as a benchmark problem the case of a harmonic potential, for which we can compare our numerical results against the analytical steady-state solution of this problem.