Dr Werner Bauer

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer, Behrens and Cotter, 2019), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

Modern high performance computers are massively parallel; for many PDE applications spatial parallelism saturates long before the computer's capability is reached. Parallel-in-time methods enable further speedup beyond spatial saturation by solving multiple timesteps simultaneously to expose additional parallelism. ParaDiag is a particular approach to parallel-in-time based on preconditioning the simultaneous timestep system with a perturbation that allows block diagonalisation via a Fourier transform in time. In this article, we introduce asQ, a new library for implementing ParaDiag parallel-in-time methods, with a focus on applications in the geosciences, especially weather and climate. asQ is built on Firedrake, a library for the automated solution of finite element models, and the PETSc library of scalable linear and nonlinear solvers. This enables asQ to build ParaDiag solvers for general finite element models and provide a range of solution strategies, making testing a wide array of problems straightforward. We use a quasi-Newton formulation that encompasses a range of ParaDiag methods, and expose building blocks for constructing more complex methods. The performance and flexibility of asQ is demonstrated on a hierarchy of linear and nonlinear atmospheric flow models. We show that ParaDiag can offer promising speedups and that asQ is a productive testbed for further developing these methods.

We introduce a physically relevant stochastic representation of the rotating shallow water equations. The derivation relies mainly on a stochastic transport principle and on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. As for the classical (deterministic) system, this scheme, referred to as modelling under location uncertainty (LU), conserves the global energy of any realization and provides the possibility to generate an ensemble of physically relevant random simulations with a good trade-off between the model error representation and the ensemble's spread. To maintain numerically the energy conservation feature, we combine an energy (in space) preserving discretization of the underlying deterministic model with approximations of the stochastic terms that are based on standard finite volume/difference operators. The LU derivation, built from the very same conservation principles as the usual geophysical models, together with the numerical scheme proposed can be directly used in existing dynamical cores of global numerical weather prediction models. The capabilities of the proposed framework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on the sphere.

Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

We introduce a geometric variational discretization framework for geophysical flow models. The numerical scheme is obtained by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Being based on a discrete version of the Euler-Poincare variational method, this discretization approach is widely applicable. We present an overview of structure-preserving variational discretizations of various equations of geophysical fluid dynamics, such as the Boussinesq, anelastic, pseudo-incompressible, and rotating shallow-water equations. We verify the structure-preserving nature of the resulting variational integrators for test cases of geophysical relevance. Our framework applies to irregular mesh discretizations in 2D and 3D in planar and spherical geometry and produces schemes that preserve invariants of the equations such as mass and potential vorticity. Descending from variational principles, the discussed variational schemes exhibit a discrete version of Kelvin circulation theorem and show excellent long term energy behavior.

We introduce a higher order phase averaging method for nonlinear oscillatory systems. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. Phase averaging is useful for deriving reduced models that can be solved numerically with more efficiency, since larger timesteps can be taken. Recently, Haut and Wingate (2014) introduced the idea of computing finite window numerical phase averages in parallel as the basis for a coarse propagator for a parallel-in-time algorithm. In this contribution, we provide a framework for higher order phase averages that aims to better approximate the unaveraged system whilst still filtering fast motions. Whilst the basic phase average assumes that the solution independent of changes of phase, the higher order method expands the phase dependency in a basis which the equations are projected onto. In this new framework, the original numerical phase averaging formulation arises as the lowest order version of this expansion. Our new projection onto functions that are $k$th degree polynomials in the phase gives rise to higher order corrections to the phase averaging formulation. We illustrate the properties of this method on an ODE that describes the dynamics of a swinging spring due to Lynch (2002). Although idealized, this model shows an interesting analogy to geophysical flows as it exhibits a slow dynamics that arises through the resonance between fast oscillations. On this example, we show convergence to the non-averaged (exact) solution with increasing approximation order also for finite averaging windows. At zeroth order, our method coincides with a standard phase average, but at higher order it is more accurate in the sense that solutions of the phase averaged model track the solutions of the unaveraged equations more accurately.

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.

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincar\'{e} reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincar\'{e} equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and the excellent conservational properties of the discrete variational integrator.

This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

The anelastic and pseudo-incompressible equations are two wellknown soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincare variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighted velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases.

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincar\'e variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighed velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases.

We describe an energy-enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy-enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy-enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere.

We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric-dependent closure equations that apply the Hodge-star operator. Our discretization framework conserves this geometrical structure and provides for all DFs proper FE spaces such that the differential operators hold in strong form. We introduce lowest possible order discretizations of the split 1D wave equations, in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant FE spaces, omitting partial integrations. Approximating the Hodge-star by nontrivial Galerkin projections (GP), the two discrete metric equations follow by projections onto either the piecewise constant (GP0) or piecewise linear (GP1) space. Our framework gives us three schemes with significantly different behavior. The split scheme using twice GP1 is unstable and shares the dispersion relation with the P1-P1 FE scheme that approximates both variables by piecewise linear spaces (P1). The split schemes that apply a mixture of GP1 and GP0 share the dispersion relation with the stable P1-P0 FE scheme that applies piecewise linear and piecewise constant (P0) spaces. However, the split schemes exhibit second order convergence for both quantities of interest. For the split scheme applying twice GP0, we are not aware of a corresponding standard formulation to compare with. Though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast, the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find FE schemes by new combinations of FE spaces.

WG3 discussed both the pros and cons of existing schemes as Working group 1 considered the treatment of model uncertainty (MU) in high-resolution ensembles, at grid spacings of order 1-5 km. These systems are often run for regional weather forecasting, perhaps over a single country, and for lead times of up to 5 days. Looking ahead, ECMWF’s strategy seeks to deliver global medium-range ensemble forecasts with 3-4 km grid spacings by 2030. It is questionable for what grid spacing we should dispense with a deep convection parameterization, but it will be either switched off or damped in these systems, such that deep convection can be assumed to be dominated by explicit motions. One of the problems with limited-area ensemble systems at this scale is that spread depends not only on the modelling system itself but also on the variability inherited from the large-scale boundary conditions. There is often thought to be a lack of spread in our high-resolution EPS (ensemble prediction systems), but this could reflect a lack of diversity on larger scales. The relative importance of lateral-boundary diversity and the model uncertainty mechanisms is regime dependent. The lateral boundaries will generally be more important in midlatitude winter but less so for summertime convection in relatively weak synoptic flow.

We describe an energy-enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy-enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy-enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere. (C) 2018 The Authors. Published by Elsevier Inc.

In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg’s principle, we overcame these difficulties by blending Koopman’s classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz that introduces the trajectories of computational particles—the koopmons—sampling the Lagrangian classical paths in phase space. In the case of Tully’s nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results.

We introduce a structure preserving discretization of stochastic rotating shallow water equations, stabilized with an energy conserving Casimir (i.e. potential enstrophy) dissipation. A stabilization of a stochastic scheme is usually required as, by modeling subgrid effects via stochastic processes, small scale features are injected which often lead to noise on the grid scale and numerical instability. Such noise is usually dissipated with a standard diffusion via a Laplacian which necessarily also dissipates energy. In this contribution we study the effects of using an energy preserving selective Casimir dissipation method compared to diffusion via a Laplacian. For both, we analyze stability and accuracy of the stochastic scheme. The results for a test case of a barotropically unstable jet show that Casimir dissipation allows for stable simulations that preserve energy and exhibit more dynamics than comparable runs that use a Laplacian.

We introduce a structure preserving discretization of stochastic rotating shallow water equations, stabilized with an energy conserving Casimir (i.e. potential enstrophy) dissipation. A stabilization of a stochastic scheme is usually required as, by modeling subgrid effects via stochastic processes, small scale features are injected which often lead to noise on the grid scale and numerical instability. Such noise is usually dissipated with a standard diffusion via a Laplacian which necessarily also dissipates energy. In this contribution we study the effects of using an energy preserving selective Casimir dissipation method compared to diffusion via a Laplacian. For both, we analyze stability and accuracy of the stochastic scheme. The results for a test case of a barotropically unstable jet show that Casimir dissipation allows for stable simulations that preserve energy and exhibit more dynamics than comparable runs that use a Laplacian.

We demonstrate how efficient r-adapted grids for the prediction of tropical cyclone (TC) tracks can be constructed with the help of goal-oriented error estimates. The binary interaction of TCs in a barotropic model is used as a test case. We perform a linear sensitivity analysis for this problem to evaluate the contribution of each grid cell to an error measure correlated with the cyclone positions. This information allows us to estimate the local grid resolution required to minimize the TC position error. An algorithm involving the solution of a Poisson problem is employed to compute how grid points should be moved such that the desired local resolution is achieved. A hexagonal shallow-water version of the next-generation numerical weather prediction and climate model ICON is used to perform model runs on these adapted grids. The results show that for adequately chosen grid adaptation parameters, the accuracy of the track prediction can be maintained even when a coarser grid is used in regions for which the estimated error contribution is low. Accurate track predictions are obtained only when a grid with high resolution consisting of cells with nearly constant size and regular shape covers the part of the domain where the estimated error contribution is large. The number of grid points required to achieve a certain accuracy in the track prediction can be decreased substantially with our approach.

We introduce a physically relevant stochastic representation of the rotating shallow water equations. The derivation relies mainly on a stochastic transport principle and on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. As for the classical (deterministic) system, this scheme, referred to as modeling under location uncertainty (LU), conserves the global energy of any realization and provides the possibility to generate an ensemble of physically relevant random simulations with a good trade-off between the model error representation and the ensemble's spread. To maintain numerically the energy conservation feature, we combine an energy (in space) preserving discretization of the underlying deterministic model with approximations of the stochastic terms that are based on standard finite volume/difference operators. The LU derivation, built from the very same conservation principles as the usual geophysical models, together with the numerical scheme proposed can be directly used in existing dynamical cores of global numerical weather prediction models. The capabilities of the proposed framework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on the sphere.

We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric-dependent closure equations that apply the Hodge-star operator. Our discretization framework conserves this geometrical structure and provides for all DFs proper FE spaces such that the differential operators (here gradient and divergence) hold in strong form. We introduce lowest possible order discretizations of the split 1D wave equations, in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant FE spaces, omitting partial integrations. Approximating the Hodge-star by nontrivial Galerkin projections (GP), the two discrete metric equations follow by projections onto either the piecewise constant (GP0) or piecewise linear (GP1) space. Out of the four possible realizations, our framework gives us three schemes with significantly different behavior. The split scheme using twice GP1 is unstable and shares the dispersion relation with the P1-P1 FE scheme that approximates both variables by piecewise linear spaces (P1). The split schemes that apply a mixture of GP1 and GP0 share the dispersion relation with the stable P1-P0 FE scheme that applies piecewise linear and piecewise constant (P0) spaces. However, the split schemes exhibit second order convergence for both quantities of interest. For the split scheme applying twice GP0, we are not aware of a corresponding standard formulation to compare with. Though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast, the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find FE schemes by new combinations of FE spaces. (C) 2017 The Authors. Published by Elsevier Inc.

•Combine compatible finite element method with Hamiltonian framework.•Include upwinding schemes while maintaining energy conservation.•Obtain more stable fields, with energy conservation up to machine precision. 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.

A stochastic representation based on a physical transport principle is proposed to account for mesoscale eddy effects on the evolution of the large-scale flow. This framework arises from a decomposition of the Lagrangian velocity into a smooth (in time) component and a highly oscillating term. One important characteristic of this random model is that it conserves the energy of any transported scalar. Such an energy -preserving representation is tested for the coarse simulation of a barotropic circulation in a shallow ocean basin, driven by a symmetric double -gyres wind forcing. The empirical spatial correlation of the random small-scale velocity is estimated from data of an eddy -resolving simulation. After reaching a turbulent equilibrium state, a statistical analysis of tracers shows that the proposed random model enables us to reproduce accurately, on a coarse mesh, the local structures of the first four statistical moments (mean, variance, skewness and kurtosis) of the high -resolution eddy -resolved data.

An important open question in fluid dynamics concerns the effect of small-scales in structuring a fluid flow. In oceanic or atmospheric flows, this is aptly captured in wave-current interactions through the study of the well-known Langmuir secondary circulation. Such wave-current interactions are described by the Craik-Leibovich system, in which the action of a wave induced velocity, the Stokes drift, produces a so called "vortex force" that causes streaking in the flow. In this work, we show that these results can be generalized as a generic effect of the spatial inhomogeneity of the statistical properties of the small-scale flow components. As demonstrated, this is well captured through a stochastic representation of the flow.

We introduce n-dimensional equations of geophysical fluid dynamics (GFD) valid on rotating n-dimensional manifolds. Moreover, by using straight and twisted differential forms and an auxiliary velocity field, we introduce hierarchically-structured equations of (geophysical) fluid dynamics in which the equations are split into metric-free and metric-dependent parts. For these sets of equations we provide representations in local coordinate charts and we show that they conserve potential vorticity and that Kelvin's circulation theorem holds. As such general n-dimensional formulations do not exist in vector calculus, we provide for both covariant and vector-invariant equations a representation on a rotating coordinate frame in an Euclidean space and compare these representations. This study reveals, among others, that the prognostic variables, described by straight and twisted differential forms, are independent of both metric and orientation. This makes them perfect descriptors of the fluid's quantities of interest, as they assign, analogously to physical measurement devices, real valued numbers to finite distances, areas, or volumes. This is not the case for prognostic variables described by (vector) proxy fields, as they depend on metric and orientation. The new structuring reveals also important geometrical features of the equations of GFD. For instance, the dimensioned differential forms display the geometric nature of the fluid's characteristics, while the equations' structuring illustrates how the metric-free momentum and continuity equations geometrically interact and how they are connected by the metric-dependent Hodge star and Riemannian lift. Besides this geometric insight, this structuring has also practical benefits: since equations containing only topological structure are less complicated to implement and can be integrated exactly, our results may contribute to more efficient and accurate discretizations.

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework [1,2], we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

We present an energy conserving space discretisation based on a Poisson bracket that can be used to derive the dry compressible Euler as well as thermal shallow water equations. It is formulated using the compatible finite element method, and extends the incorporation of upwinding for the shallow water equations as described in Wimmer, Cotter, and Bauer (2019). While the former is restricted to DG upwinding, an energy conserving SUPG scheme for the (partially) continuous Galerkin thermal field space is newly introduced here. The energy conserving property is validated by coupling the Poisson bracket based spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved temperature field development with respect to stability when upwinding is included. An approximately energy conserving full discretisation with a smaller computational cost is also presented.

TRiSK-type numerical schemes are widely used in both atmospheric and oceanic dynamical cores, due to their discrete analogues of important properties such as energy conservation and steady geostrophic modes. In this work, we show that these numerical methods are best understood as a discrete exterior calculus (DEC) scheme applied to a Hamiltonian formulation of the rotating shallow water equations based on split exterior calculus. This comprehensive description of the differential geometric underpinnings of TRiSK-type schemes completes the one started in \cite{Thuburn2012,Eldred2017}, and provides a new understanding of certain operators in TRiSK-type schemes as discrete wedge products and topological pairings from split exterior calculus. All known TRiSK-type schemes in the literature are shown to fit inside this general framework, by identifying the (implicit) choices made for various DEC operators by the different schemes. In doing so, unexplored choices and combinations are identified that might offer the possibility of fixing known issues with TRiSK-type schemes such as operator accuracy and Hollingsworth instability.

This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincare reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincare equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and excellent conservational properties of the discrete variational integrator.

Numerical models of weather and climate critically depend on the long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2), 495 (2013)], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [Pavlov et al., "Structure-preserving discretization of incompressible fluids, " Physica D 240(6), 443-458 (2011)] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.