Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

Abstract

We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.

arXiv comment18 pages, 8 figures, first version: all comments welcome

References in corpus (11)

• Firedrake: automating the finite element method by composing abstractions
• Mixed finite elements for numerical weather prediction
• TSFC: a structure-preserving form compiler
• Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements
• Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
• Compatible finite element spaces for geophysical fluid dynamics
• Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions
• A variational H(div) finite element discretisation approach for perfect incompressible fluids
• An algorithm for the optimization of finite element integration loops
• A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere
• Scale-selective dissipation in energy-conserving finite element schemes for two-dimensional turbulence

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• A Mixed Mimetic Spectral Element Model of the 3D Compressible Euler Equations on the Cubed Sphere
• An energetically balanced, quasi-Newton integrator for non-hydrostatic vertical atmospheric dynamics
• Petrov-Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems
• Energy preserving reduced-order modelling of thermal shallow water equation
• Structure-preserving Reduced Order Modeling of non-traditional Shallow Water Equation
• Selective decay for the rotating shallow-water equations with a structure-preserving discretization