We investigate a degenerate, linear, periodic advection–diffusion equation in one dimension. The problem is hyperbolic in a subinterval and parabolic in the complement with boundary conditions that impose periodicity of the advective–diffusive flux to ensure mass conservation. To select the 'physically acceptable' solution as the limit of vanishing viscosity solutions, continuity of the solution at the parabolic to hyperbolic interface is imposed, following Gastaldi and Quarteroni (1989). Using this condition, we establish the well-posedness of the Cauchy problem in the framework of the evolution linear semi-groups theory. We also discuss the regularity of the solution when the initial condition is too rough to be in the domain of the evolution operator. We then present reference solutions obtained using this additional interface condition. These solutions can be used to test the robustness of numerical schemes. Finally, we discuss the numerical results of an upwind scheme, a finite volume box-scheme, and a local discontinuous Galerkin method. These three schemes, which do not explicitly enforce the additional interface condition, automatically select the physically acceptable solution, with the two latter schemes being more accurate.