The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by \cite{AUG84} who discretized Hamilton's action principle with a free-Lagrange data structure. The purpose of this article is to assess the long-time conservation properties of the VFL method for regularized shallow water which are useful for climate simulation. Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which satisfy semi-discrete divergence and potential vorticity conservation laws. These diagnostic semi-discrete equations augment the description of the VFL method by characterizing the evolution of its respective irrotational and solenoidal components in the Lagrangian frame. Like the continuum equations, the former exhibits a $\text{div}^2\mathbf{U}$ term which indicates that the flow has a very strong tendency towards a purely rotational state.
Numerical results show (i) the preservation of shape and strength of an initially radially symmetric vortex pair in purely rotational regularized shallow water and (ii) how the Voronoi diagram retains the history of the flow field and (iii) that energy is conserved to $\mathcal{O}(Δ^2)$ and potential vorticity error to within 5% with no secular growth over a 50 year period.
A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay（凡尔赛数学实验室，凡尔赛大学，法国国家科学研究中心，巴黎萨克雷大学）；Maison de la Simulation USR 3441, Digiteo Labs, bât. 565, PC 190, CEA Saclay（模拟之家 USR 3441，Digiteo 实验室，CEA 萨克雷）；CEA/DEN/DANS/DM2S/STMF - CEA Saclay（CEA/DEN/DANS/DM2S/STMF - CEA 萨克雷）
A two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition and its application to the Bay of Bengal region. Part I: Flat bottom topography
Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation
A high-order velocity-based discontinuous Galerkin scheme for the shallow water equations: local conservation, entropy stability, well-balanced property, and positivity preservation
High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography
On Hyperbolic Stochastic Galerkin Projections of Shallow Water Linearised Moment Equations
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；CogniGron (Groningen Cognitive Systems and Materials Centre), University of Groningen（格罗宁根大学认知系统材料中心（CogniGron））；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学与统计系）
The Riemann problem for the shallow water equations with discontinuous topography
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université de Paris 6（雅克-路易·利翁斯实验室与法国国家科学研究中心，巴黎第六大学）；Department of Mathematics, International University（国际大学数学系）
A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室 & 法国国家科学研究中心 皮埃尔和玛丽·居里大学）；Department of Mathematics, International University（国际大学数学系）
A moment model of shallow granular flows with variable friction laws
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学和统计系）
Entropy analysis and entropy stable DG methods for the 1D shallow water moment equations
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；Department of Mathematics, Linköping University（林雪平大学数学系）；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学和统计系）
