Finite Volume Approximation of One-Dimensional Stiff Convection-Diffusion Equations

In this work, we present a novel method to approximate stiff problems using a finite volume (FV) discretization. The stiffness is caused by the existence of a small parameter in the equation which introduces a boundary layer. The proposed semi-analytic method consists in adding in the finite volume space the boundary layer corrector which encompasses the singularities of the problem. We verify the stability and convergence of our finite volume schemes which take into account the boundary layer structures. A major feature of the proposed scheme is that it produces an efficient stable second order scheme to be compared with the usual stable upwind schemes of order one or the usual costly second order schemes demanding fine meshes.