Analysis and simulation of a coupled hyperbolic/parabolic model problem

We investigate a periodic, one-dimensional, linear, and degenerate advection-diffusion equation. The problem is hyperbolic in a subinterval and parabolic in the complement, and the boundary conditions only impose the periodicity of the advective-diffusive flux to ensure mass conservation. Following Gastaldi and Quarteroni (1989), a condition is added to select the “physically acceptable” solution as the limit of vanishing viscosity solutions, namely the continuity of the solution at the parabolic to hyperbolic interface. 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. Then, we present reference solutions obtained using the additional interface condition. These solutions can be used to test the robustness of numerical schemes. Finally, we discuss results obtained with an upwind scheme, a finite volume box-scheme, and a local discontinuous Galerkin method. The three schemes (which do not enforce explicitly the additional interface condition) select automatically the physically acceptable solution, the two latter schemes being more accurate.