From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations

Abstract

By revisiting the basic Godunov approach for system of linear hyperbolic Partial Differential Equations (PDEs) we show that it is hybridizable. As such, it is a natural recipe for us to constructively and systematically establish a unified hybridized discontinuous Galerkin (HDG) framework for a large class of PDEs including those of Friedrichs' type. The unification is fourfold. First, it provides a single constructive procedure to devise HDG schemes for elliptic, parabolic, hyperbolic, and mixed-type PDEs. The key that we exploit is the fact that, for many PDEs, irrespective of their type, the first order form is a hyperbolic system. Second, it reveals the nature of the trace unknowns as the upwind states. Third, it provides a parameter-free HDG framework, and hence eliminating the “usual complaint” that HDG is a parameter-dependent method. Fourth, it allows us to rediscover most existing HDG methods and furthermore discover new ones.We apply the proposed unified framework to three different PDEs: the convection–diffusion–reaction equation, the Maxwell equation in frequency domain, and the Stokes equation. The purpose is to present a step-by-step construction of various HDG methods, including the most economic ones with least trace unknowns, by exploiting the particular structure of the underlying PDEs. The well-posedness of the resulting HDG schemes, i.e. the existence and uniqueness of the HDG solutions, is proved. The well-posedness result is also extended and proved for abstract Friedrichs' systems. We also discuss variants of the proposed unified framework and extend them to the popular Lax–Friedrichs flux and to nonlinear PDEs. Numerical results for transport equation, convection–diffusion equation, compressible Euler equation, and shallow water equation are presented to support the unification framework.