On the transport limit of singularly perturbed convection–diffusion problems on networks

Herbert Egger,Nora Philippi

doi:10.1002/mma.7084

Abstract

We consider singularly perturbed convection–diffusion equations on one‐dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling conditions at inner vertices are derived that guarantee conservation of mass and dissipation of a mathematical energy which allows us to prove stability and well‐posedness. For single intervals and appropriately specified initial conditions, it is well‐known that the solutions of the convection–diffusion problem converge to that of the transport problem with order in the L∞(L2)‐norm with diffusion ϵ → 0. In this paper, we prove a corresponding result for problems on one‐dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.

Highlights

The transport and diffusion of a chemical substance in the stationary flow of an incompressible fluid through a pipe can be described by (1)a∂tuǫ(x, t) + b∂xuǫ(x, t) = ǫ∂xxuǫ(x, t), which is assumed to hold for x ∈ (0, l) and t > 0
Equations (1) and (3) are assumed to hold for every single pipe while the boundary conditions (2) and (4) have to be augmented by appropriate coupling conditions at pipe junctions. These have to be chosen in order to guarantee conservation of mass across network junctions as well as dissipation of a mathematical energy, which is utilized to ensure the well-posedness of the problems
The previous theorem shows that the asymptotic analysis of convection– diffusion problems can be extended almost verbatim to networks, if appropriate coupling conditions and corresponding boundary layer functions are defined at the network junctions

Summary

Introduction

The transport and diffusion of a chemical substance in the stationary flow of an incompressible fluid through a pipe can be described by (1). Equations (1) and (3) are assumed to hold for every single pipe while the boundary conditions (2) and (4) have to be augmented by appropriate coupling conditions at pipe junctions. These have to be chosen in order to guarantee conservation of mass across network junctions as well as dissipation of a mathematical energy, which is utilized to ensure the well-posedness of the problems. One of the main difficulties in the asymptotic analysis here is that the number and type of coupling conditions changes in the singular limit ǫ → 0 This gives rise to additional internal layers at pipe junctions that need to be handled appropriately.

Notation and preliminaries

Asymptotic analysis