On compatible finite elements for atmosphere modelling
File(s)
Author(s)
Kowalczyk, Karina
Type
Thesis
Abstract
Simulations of atmospheric dynamics are the foundation of numerical weather prediction and climate projections. It is crucial for numerical models of large-scale geophysical flows to capture the relevant balances exactly. Compatible finite elements have been applied successfully for geophysical flow simulations, allowing for a variety of underlying mesh structures and higher-order approximations, while maintaining desirable structure-preserving properties.
The presence of orography has long been a challenge in numerical weather prediction. Pressure gradient errors, for instance, that occur in finite-difference models as a result of using sigma-coordinates, appear also in compatible finite element discretisations on terrain-following meshes. Here, they are a consequence of the Piola transform that introduces a vertical component into the horizontal part of the velocity space. While the natural velocity space on flat meshes decomposes into horizontal and vertical parts, this is no longer the case in the presence of orography. We propose a finite element space for the fluid velocity that retains the split into horizontal and vertical components. By reformulating the discrete finite element problem, we show that this space is suitable for approximations of the governing compressible equations.
Finding a time-stepping scheme for compatible finite element atmosphere models that allows for large stable time-steps has remained an open challenge. The choice of a time-stepping scheme is critical to balancing computational efficiency, numerical stability, and accuracy in atmosphere models. The multi-scale nature of atmospheric flows poses significant challenges, with the fast wave dynamics often restricting maximum stable time-steps. We propose a scalable semi-implicit projection time-stepping scheme based on a splitting of advection and wave dynamics with time-steps constrained by the advection step only. Starting with the shallow-water equations, we show the numerical robustness of this scheme. In a second step, we provide a formulation for the full compressible equations.
The presence of orography has long been a challenge in numerical weather prediction. Pressure gradient errors, for instance, that occur in finite-difference models as a result of using sigma-coordinates, appear also in compatible finite element discretisations on terrain-following meshes. Here, they are a consequence of the Piola transform that introduces a vertical component into the horizontal part of the velocity space. While the natural velocity space on flat meshes decomposes into horizontal and vertical parts, this is no longer the case in the presence of orography. We propose a finite element space for the fluid velocity that retains the split into horizontal and vertical components. By reformulating the discrete finite element problem, we show that this space is suitable for approximations of the governing compressible equations.
Finding a time-stepping scheme for compatible finite element atmosphere models that allows for large stable time-steps has remained an open challenge. The choice of a time-stepping scheme is critical to balancing computational efficiency, numerical stability, and accuracy in atmosphere models. The multi-scale nature of atmospheric flows poses significant challenges, with the fast wave dynamics often restricting maximum stable time-steps. We propose a scalable semi-implicit projection time-stepping scheme based on a splitting of advection and wave dynamics with time-steps constrained by the advection step only. Starting with the shallow-water equations, we show the numerical robustness of this scheme. In a second step, we provide a formulation for the full compressible equations.
Version
Open Access
Date Issued
2025-06-15
Date Awarded
01/02/2026
License URL
Advisor
Cotter, Colin
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)