Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes

JOURNAL OF COMPUTATIONAL PHYSICS

卷:418

Article

Marche, Fabien¹ 

[1] Univ Montpellier, Inst Montpellierain Alexander Grothendieck, F-34095 Montpellier, France

关键词: Green-Naghdi equations;    High-order schemes;    Hybridized Discontinuous Galerkin;    Free surface flows;    Shallow water equations;    Nonlinear dispersive equations;   

DOI  :  10.1016/j.jcp.2020.109637

来源: Elsevier

【 摘 要 】

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free