We develop structure-preserving numerical methods for the Serre-Green-Naghdi equations, a model for weakly dispersive free-surface waves. We consider both the classical form, requiring the inversion of a non-linear elliptic operator, and a hyperbolic approximation of the equations, allowing fully explicit time stepping. Systems for both flat and variable topography are studied. Our novel numerical methods conserve both the total water mass and the total energy. In addition, the methods for the original Serre-Green-Naghdi equations conserve the total momentum for flat bathymetry. For variable topography, all the methods proposed are well-balanced for the lake-at-rest state. We provide a theoretical setting allowing us to construct schemes of any kind (finite difference, finite element, discontinuous Galerkin, spectral, etc.) as long as summation-by-parts operators are available in the chosen setting. Energy-stable variants are proposed by adding a consistent high-order artificial viscosity term. The proposed methods are validated through a large set of benchmarks to verify all the theoretical properties. Whenever possible, comparisons with exact, reference numerical, or experimental data are carried out. The impressive advantage of structure preservation, and in particular energy preservation, to resolve accurately dispersive wave propagation on very coarse meshes is demonstrated by several of the tests.
GPU-Accelerated Energy-Conserving Methods for the Two-Dimensional Hyperbolized Serre-Green-Naghdi Equations
Massachusetts Institute of Technology（麻省理工学院）；Johannes Gutenberg University Mainz（美因茨约翰内斯古腾堡大学）；INRIA, U. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（法国国家信息与自动化研究所、波尔多大学、法国国家科学研究中心、波尔多国立理工学院、数学与交叉研究实验室）
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
INSA Toulouse（图卢兹国立应用科学学院）；University of Montpellier, Institut Montpelliérain Alexander Grothendieck and Inria LEMON team（蒙彼利埃大学，亚历山大·格罗滕迪克蒙彼利埃研究所和Inria LEMON团队）
Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model
Université de Toulouse, UPS/INSA, IMT, CNRS UMR 5219（图卢兹大学）；DMA, Ecole Normale Supérieure, et CNRS UMR 8553（巴黎高等师范学院）；I3M, Univ. Montpellier 2 et CNRS UMR 5149（蒙彼利埃第二大学）
Discrete transparent boundary conditions for the linearized Green-Naghdi system of equations
Institut de Mathématiques de Toulouse（图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，UPS IMT）；UMR5219, Université de Toulouse（图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，INSA）；CNRS, UPS IMT；Institut de Mathématiques de Toulouse；UMR5219, Université de Toulouse；CNRS, INSA
An energy-consistent depth-averaged Euler system: derivation and properties
Inria（法国国家信息与自动化研究所）；CEREMA（法国中央城市与区域设备局）；CNRS（法国国家科学研究中心）；Laboratoire Jacques-Louis Lions（雅克-路易·利翁斯实验室）；Institut de Physique du Globe de Paris（巴黎地球物理学院）
