A Scharfetter-Gummerl stabilization scheme for HDG approximations of convection-diffusion problems

TL;DR

This paper introduces a Scharfetter-Gummel stabilization scheme for high-order HDG methods applied to convection-diffusion problems, ensuring stability and equivalence to finite volume methods in one dimension.

Contribution

The paper develops a novel stabilization scheme for high-order HDG methods, linking it to finite volume methods and enhancing stability for convection-diffusion problems.

Findings

SG-HDG scheme is equivalent to stabilized finite volume method in 1D

The stabilization parameters are carefully chosen for improved accuracy

The scheme applies to all orders of HDG methods

Abstract

We present a Scharfetter-Gummel (SG) stabilization scheme for high-order Hybrid Discontinuous Galerkin (HDG) approximations of convection-diffusion problems. The scheme is based on a careful choice of the stabilization parameters used to define the numerical flux in the HDG method. We show that, in one dimension, the SG-HDG scheme is equivalent to the Finite Volume method stabilized with the Scharfetter--Gummel on the dual grid, for all orders of HDG schemes.