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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:295

Article

Bui-Thanh, Tan

关键词: Discontinuous Galerkin methods;    Hybridized discontinuous Galerkin methods;    Upwind hybridized discontinuous Galerkin methods;    Godunov method;    Riemann flux;    Lax-Friedrichs flux;    Friedrichs' system;    Upwind;    Well-posedness;    Compressible Euler equation;    Shallow water equation;    Newton method;   

DOI  :  10.1016/j.jcp.2015.04.009

来源: Elsevier

【 摘 要 】

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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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