A mixed finite-element discretisation of the shallow-water equations

Kent, James; Melvin, Thomas; Wimmer, Golo A.

doi:10.5194/gmd-16-1265-2023

Cited by 9 publications

(13 citation statements)

References 22 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bendall and showed in numerical experiments (including a nonlinear rotating shallow water equation test case on the sphere) that this discretization produces second-order accurate solutions using lowest-order RT quadrilateral elements on a cubed sphere grid, whilst (4.19) only produces first-order accurate solutions. A predecessor of this scheme was considered in Kent, Melvin and Wimmer (2023), but without the consistent definition of and .…”

Section: Transport Of (Div) Fieldsmentioning

confidence: 99%

Compatible finite element methods for geophysical fluid dynamics

Cotter¹

2023

Acta Numerica

View full text Add to dashboard Cite

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

show abstract

Section: Transport Of (Div) Fieldsmentioning

confidence: 99%

Compatible finite element methods for geophysical fluid dynamics

Cotter¹

2023

Acta Numerica

View full text Add to dashboard Cite

show abstract

“…Here we use two explicit timesteppers: a fourth order Runge-Kutta method, RK4, and a strongly structure preserving Runge-Kutta method, SSPRK3; and an implicit midpoint time-stepper. We also use a time-stepping scheme similar to the Iterated Semi-Implicit (ISI) approach (Kent et al, 2023;Melvin et al, 2019) used in LFRic. Within Gusto this is refered to as the 'Semi-Implicit Quasi-Newton' (SIQN) scheme.…”

Section: Gustomentioning

confidence: 99%

“…LFRic is the next generation weather and climate model developed by the MetOffice; a detailed overview of the shallow water component of the model is found in (Kent et al, 2023). Its newly developed dynamical core, 'GungHo' (Melvin et al, 2019), uses a compatible finite element discretisation on quadrilateral elements.…”

Section: Lfricmentioning

confidence: 99%

“…The finite element function spaces are RT 1 − Q0. LFRic's default time-stepper is an Iterated Semi-Implicit (ISI) approach (Kent et al, 2023;Melvin et al, 2019). Two explicit time-steppers, SSPRK3 and RK4, are compared to this.…”

Section: Lfricmentioning

confidence: 99%

“…We use the Rotating Shallow Water Equations (RSWEs) as a simple model for fluids in the oceans and atmosphere. These are well-suited for analysing numerical discretisations, being a reduced equation set that retains the key slow and fast wave dynamics of the full equations (Aechtner et al, 2015;Kent et al, 2023;Ullrich et al, 2010;Ullrich et al, 2014). The RSWEs demonstrate that nonlinearities have an important contribution over a longer time-period.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantifying the accuracy of large time-steps in highly oscillatory systems

Andrews¹,

Shipton²

2023

Preprint

View full text Add to dashboard Cite

<p>There is an ever-increasing demand for longer and higher resolution numerical simulations of the weather and climate. To achieve this with reasonable wall-clock times, it is desirable to use as large a time-step as possible, whilst retaining a stable and accurate solution. However, this is very challenging in the presence of highly oscillatory linear waves; there are explicit time-step limits and losses in accuracy with implicit methods. This talk will highlight where non-linear errors can result from large time-steps and provide metrics for quantifying this. We begin by re-casting the non-linearity as a product of linear waves. In the Rotating Shallow Water Equations (RSWEs), this allows for key dynamics to be expressed as three-wave &#8216;triad&#8217; interactions. A non-linear &#8216;triadic&#8217; time-stepping error is computed using linear stability polynomials. A number of explicit and implicit time-stepping methods (such as RK4, TR-BDF2, ETD-RK2) will be compared analytically in the RSWEs. Next, two new test problems enable analyses of large time-step simulations. The first is of a Gaussian perturbation to a RSWE height field. A proposed metric, relating to the kinetic energy distribution over temporal frequency, quantifies phase errors in the height reformation. The second test case initialises linear waves which, via direct- and near- resonant triad interactions, will construct non-linear dynamics. Phase errors with large time-steps can be identified in the corresponding height fields. A first variant of this case will initialise only two waves; this will primarily instigate an energy exchange within a dominant triad. A second version, containing more slow modes, enables a re-distribution of fast mode energy into rings in wavenumber space.</p>

show abstract

A mixed finite‐element, finite‐volume, semi‐implicit discretisation for atmospheric dynamics: Spherical geometry

Melvin,

Shipway,

Wood

et al. 2024

Quart J Royal Meteoro Soc

View full text Add to dashboard Cite

Our previously described reformulation of the Met Office's dynamical core for weather and climate prediction is extended to spherical domains using a cubed‐sphere mesh. We update the semi‐implicit mixed finite‐element formulation to be suitable for spherical domains. In particular, the finite‐volume transport scheme is extended to take account of non‐uniform, non‐orthogonal meshes and uses an advective‐then‐flux formulation so that increment from the transport scheme is linear in the divergence. The resulting model is then applied to a standard set of dry dynamical core tests and compared with the existing semi‐implicit semi‐Lagrangian dynamical core currently used in the Met Office's operational model.

show abstract

A mixed finite-element discretisation of the shallow-water equations

Cited by 9 publications

References 22 publications

Compatible finite element methods for geophysical fluid dynamics

Compatible finite element methods for geophysical fluid dynamics

Quantifying the accuracy of large time-steps in highly oscillatory systems

A mixed finite‐element, finite‐volume, semi‐implicit discretisation for atmospheric dynamics: Spherical geometry

Contact Info

Product

Resources

About