Volume 17, Issue 3
Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Arnaud Duran & Fabien Marche

Commun. Comput. Phys., 17 (2015), pp. 721-760.

Published online: 2018-04

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  • Abstract

We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilizing the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

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@Article{CiCP-17-721, author = {}, title = {Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {17}, number = {3}, pages = {721--760}, abstract = {

We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilizing the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.150414.101014a}, url = {http://global-sci.org/intro/article_detail/cicp/10975.html} }
TY - JOUR T1 - Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations JO - Communications in Computational Physics VL - 3 SP - 721 EP - 760 PY - 2018 DA - 2018/04 SN - 17 DO - http://doi.org/10.4208/cicp.150414.101014a UR - https://global-sci.org/intro/article_detail/cicp/10975.html KW - AB -

We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilizing the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

Arnaud Duran & Fabien Marche. (2020). Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations. Communications in Computational Physics. 17 (3). 721-760. doi:10.4208/cicp.150414.101014a
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