From Rankine-Hugoniot Condition to a Constructive Derivation of HDG Methods

This chapter presents a constructive derivation of HDG methods for convection-diffusion-reaction equation using the Rankine-Hugoniot condition. This is possible due to the fact that, in the first order form, convection-diffusion-reaction equation is a hyperbolic system. As such it can be discretized using the standard upwind DG method. The key is to realize that the Rankine-Hugoniot condition naturally provides an upwind HDG framework. The chief idea is to first break the uniqueness of the upwind flux across element boundaries by introducing single-valued new trace unknowns on the mesh skeleton, and then re-enforce the uniqueness via algebraic conservation constraints. Essentially, the HDG framework is a redesign of the standard DG approach to reduce the number of coupled unknowns. In this work, an upwind HDG method with one trace unknown is systematically constructed, and then extended to a family of penalty HDG schemes. Various existing HDG methods are rediscovered using the proposed framework.