An asymptotic preserving hybrid-dG method for convection-diffusion equations on pipe networks

TL;DR

This paper introduces an asymptotic-preserving hybrid discontinuous Galerkin method for accurately approximating convection-diffusion equations on pipe networks, effectively handling singularities and boundary layers in the vanishing diffusion limit.

Contribution

The paper develops a novel hybrid-dG scheme that automatically manages variable boundary and coupling conditions, providing uniform error estimates in the singular perturbation limit.

Findings

The method is asymptotic-preserving and handles boundary layers effectively.

Numerical tests confirm optimal error estimates.

The scheme adapts to variable conditions at network junctions.

Abstract

We study the numerical approximation of singularly perturbed convection-diffusion problems on one-dimensional pipe networks. In the vanishing diffusion limit, the number and type of boundary conditions and coupling conditions at network junctions changes, which gives rise to singular layers at the outflow boundaries of the pipes. A hybrid discontinuous Galerkin method is proposed, which provides a natural upwind mechanism for the convection-dominated case. Moreover, the method automatically handles the variable coupling and boundary conditions in the vanishing diffusion limit, leading to an asymptotic-preserving scheme. A detailed analysis of the singularities of the solution and the discretization error is presented, and an adaptive strategy is proposed, leading to order optimal error estimates that hold uniformly in the singular perturbation limit. The theoretical results are confirmed by numerical tests.