A Kinematic Conservation Law in Free Surface Flow

The Green-Naghdi system is used to model highly nonlinear weakly dispersive waves propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It is shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green-Naghdi theory which turns out to be identical to the fourth conservation law for this system.Comment: 15 pages, 1 figur

A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies

A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system

A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results

Vertically averaged and moment equations: new derivation, efficient numerical solution and comparison with other physical approximations for modeling non-hydrostatic free surface flows

Efficient modeling of flow physics is a prerequisite for a reliable computation of free-surface environmental flows. Non-hydrostatic flows are often present in shallow water environments, making the task challenging. In this work, we use the method of weighted residuals for modeling non-hydrostatic free surface flows in a depth-averaged framework. In particular, we focus on the Vertically Averaged and Moment (VAM) equations model. First, a new derivation of the model is presented using expansions of the field variables in sigma-coordinates with Legendre polynomials basis. Second, an efficient two-step numerical scheme is proposed: the first step corresponds to solving the hyperbolic part with a second-order path-conservative PVM scheme. Then, in a second step, non-hydrostatic terms are corrected by solving a linear Poisson-like system using an iterative method, thereby resulting in an accurate and efficient algorithm. The computational effort is similar to the one required for the well-known Serre-Green-Naghdi (SGN) system, while the results are largely improved. Finally, the physical aspects of the model are compared to the SGN system and a multilayer model, demonstrating that VAM is comparable in physical accuracy to a two-layer model.Funding for open access charge: Universidad de Málaga/CBUA. This work is partially supported by projects RTI2018-096064-B-C2(1-2), PID2020-114688RB-I00, and PID2022-137637NB-C21 funded by Ministry of Science, Innovation and Universities MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”. F. Cantero-Chinchilla was partially supported by the grant IJC2020-042646-I, funded by CIN/AEI/10.13039/501100011033 and by the European Union “NextGenerationEU/PRTR”, through the Spanish Ministry of Science, Innovation and Universities Juan de la Cierva program 2020.

Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

International audienceThis work is devoted to the numerical resolution in multidimensional framework of a hierarchy of reduced models of the free surface Euler equations, also called water waves equations.The current paper, the first in a series of two, focuses on a hierarchy of monolayer dispersive models, such is the Serre-Green-Naghdi model.A particular attention is given to the dissipation of the mechanical energy at the discrete level, i.e. to design an entropy-satisfying scheme.To illustrate the accuracy and the robustness of the strategy, several numerical experiments are performed.In particular, the strategy is able to deal with dry areas without particular treatment

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

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 nonlin-ear 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

Stationary shock-like transition fronts in dispersive systems

International audienceWe show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine–Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not travelling waves and apparently require for their description more complex anzats, evolve into travelling wave type jump discontinuities

Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green–Naghdi Equations

The study of finite element methods for the numerical solution of differential equations is one of the gems of modern mathematics, boasting rigorous analytical foundations as well as unambiguously useful scientific applications. Over the past twenty years, several researchers in scientific computing have realized that concepts from homological algebra and differential topology play a vital role in the theory of finite element methods. Finite element exterior calculus is a theoretical framework created to clarify some of the relationships between finite elements, algebra, geometry, and topology. The goal of this thesis is to provide an introduction to the theory of finite element exterior calculus, and to illustrate some applications of this theory to the design of mixed finite element methods for problems in geophysical fluid dynamics. The presentation is divided into two parts. Part 1 is intended to serve as a self–contained introduction to finite element exterior calculus, with particular emphasis on its topological aspects. Starting from the basics of calculus on manifolds, I go on to describe Sobolev spaces of differential forms and the general theory of Hilbert complexes. Then, I explain how the notion of cohomology connects Hilbert complexes to topology. From there, I discuss the construction of finite element spaces and the proof that special choices of finite element spaces can be used to ensure that the cohomological properties of a particular problem are preserved during discretization. In Part 2, finite element exterior calculus is applied to derive mixed finite element methods for the Green–Naghdi equations (GN). The GN extend the more well–known shallow water equations to the regime of non–infinitesimal aspect ratio, thus allowing for some non–hydrostatic effects. I prove that, using the mixed formulation of the linearized GN, approximations of balanced flows remain steady. Additionally, one of the finite element methods presented for the fully nonlinear GN provably conserves mass, vorticity, and energy at the semi–discrete level. Several computational test cases are presented to assess the practical performance of the numerical methods, including a collision between solitary waves, the motion of solitary waves over variable bottom topography, and the breakdown of an unstable balanced state