Articles | Volume 6, issue 6
https://doi.org/10.5194/gmd-6-2099-2013
© Author(s) 2013. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/gmd-6-2099-2013
© Author(s) 2013. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Development and technical paper
| 
17 Dec 2013
Development and technical paper |
| 17 Dec 2013
Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2
M. E. Rognes
Center for Biomedical Computing, Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway
D. A. Ham
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Department of Computing, Imperial College London, London SW7 2AZ, UK
C. J. Cotter
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
A. T. T. McRae
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
The Grantham Institute for Climate Change, Imperial College London, London SW7 2AZ, UK
Related authors
No articles found.
Joshua Hope-Collins, Abdalaziz Hamdan, Werner Bauer, Lawrence Mitchell, and Colin Cotter
Geosci. Model Dev., 18, 4535–4569, https://doi.org/10.5194/gmd-18-4535-2025, https://doi.org/10.5194/gmd-18-4535-2025, 2025
Short summary
Short summary
Effectively using modern supercomputers requires massively parallel algorithms. Time-parallel algorithms calculate the system state (e.g. the atmosphere) at multiple times simultaneously and have exciting potential but are tricky to implement and still require development. We have developed software to simplify implementing and testing the ParaDiag algorithm on supercomputers. We show that for some atmospheric problems it can enable faster or more accurate solutions than traditional techniques.
Reuben W. Nixon-Hill, Daniel Shapero, Colin J. Cotter, and David A. Ham
Geosci. Model Dev., 17, 5369–5386, https://doi.org/10.5194/gmd-17-5369-2024, https://doi.org/10.5194/gmd-17-5369-2024, 2024
Short summary
Short summary
Scientists often use models to study complex processes, like the movement of ice sheets, and compare them to measurements for estimating quantities that are hard to measure. We highlight an approach that ensures accurate results from point data sources (e.g. height measurements) by evaluating the numerical solution at true point locations. This method improves accuracy, aids communication between scientists, and is well-suited for integration with specialised software that automates processes.
Sia Ghelichkhan, Angus Gibson, D. Rhodri Davies, Stephan C. Kramer, and David A. Ham
Geosci. Model Dev., 17, 5057–5086, https://doi.org/10.5194/gmd-17-5057-2024, https://doi.org/10.5194/gmd-17-5057-2024, 2024
Short summary
Short summary
We introduce the Geoscientific ADjoint Optimisation PlaTform (G-ADOPT), designed for inverse modelling of Earth system processes, with an initial focus on mantle dynamics. G-ADOPT is built upon Firedrake, Dolfin-Adjoint and the Rapid Optimisation Library, which work together to optimise models using an adjoint method, aligning them with seismic and geologic datasets. We demonstrate G-ADOPT's ability to reconstruct mantle evolution and thus be a powerful tool in geosciences.
Mariana C. A. Clare, Tim W. B. Leijnse, Robert T. McCall, Ferdinand L. M. Diermanse, Colin J. Cotter, and Matthew D. Piggott
Nat. Hazards Earth Syst. Sci., 22, 2491–2515, https://doi.org/10.5194/nhess-22-2491-2022, https://doi.org/10.5194/nhess-22-2491-2022, 2022
Short summary
Short summary
Assessing uncertainty is computationally expensive because it requires multiple runs of expensive models. We take the novel approach of assessing uncertainty from coastal flooding using a multilevel multifidelity (MLMF) method which combines the efficiency of less accurate models with the accuracy of more expensive models at different resolutions. This significantly reduces the computational cost but maintains accuracy, making previously unfeasible real-world studies possible.
Thomas H. Gibson, Lawrence Mitchell, David A. Ham, and Colin J. Cotter
Geosci. Model Dev., 13, 735–761, https://doi.org/10.5194/gmd-13-735-2020, https://doi.org/10.5194/gmd-13-735-2020, 2020
Short summary
Short summary
Galerkin finite element discretizations for atmospheric modeling often require the solution of ill-conditioned, saddle point equations which can be efficiently solved using a hybridized method. By extending Firedrake's domain-specific abstraction, we provide a mechanism for the rapid implementation of hybridization methods for a wide class of methods. In this paper, we show that hybridization is an effective alternative to traditional block solvers for simulating geophysical flows.
Tuomas Kärnä, Stephan C. Kramer, Lawrence Mitchell, David A. Ham, Matthew D. Piggott, and António M. Baptista
Geosci. Model Dev., 11, 4359–4382, https://doi.org/10.5194/gmd-11-4359-2018, https://doi.org/10.5194/gmd-11-4359-2018, 2018
Short summary
Short summary
Unstructured meshes are attractive for coastal ocean modeling, as they allow more accurate representation of complex coastal topography. Unstructured mesh models are, however, often perceived as slow and inaccurate. We present a novel discontinuous Galerkin ocean model: Thetis. We demonstrate that the model is able to simulate baroclinic ocean flows with high accuracy on a triangular prismatic mesh. This work paves the way for highly accurate and efficient three-dimensional coastal ocean models.
Gheorghe-Teodor Bercea, Andrew T. T. McRae, David A. Ham, Lawrence Mitchell, Florian Rathgeber, Luigi Nardi, Fabio Luporini, and Paul H. J. Kelly
Geosci. Model Dev., 9, 3803–3815, https://doi.org/10.5194/gmd-9-3803-2016, https://doi.org/10.5194/gmd-9-3803-2016, 2016
Short summary
Short summary
Unstructured meshes offer flexibility but are perceived as slow. Some applications, including atmosphere or ocean simulations, admit an extruded mesh: the horizontal mesh may be unstructured, but the vertical dimension can be traversed in a structured way. By extending the Firedrake automated simulation framework to this case, we show that an extruded mesh can be traversed as fast as a structured mesh. This paves the way for highly efficient unstructured mesh models of the ocean and atmosphere.
J. Thuburn, C. J. Cotter, and T. Dubos
Geosci. Model Dev., 7, 909–929, https://doi.org/10.5194/gmd-7-909-2014, https://doi.org/10.5194/gmd-7-909-2014, 2014
J. Hill, E. E. Popova, D. A. Ham, M. D. Piggott, and M. Srokosz
Ocean Sci., 10, 323–343, https://doi.org/10.5194/os-10-323-2014, https://doi.org/10.5194/os-10-323-2014, 2014
Related subject area
Numerical methods
asQ: parallel-in-time finite element simulations using ParaDiag for geoscientific models and beyond
Optimized step size control within the Rosenbrock solvers for stiff chemical ordinary differential equation systems in KPP version 2.2.3_rs4
Potential-based thermodynamics with consistent conservative cascade transport for implicit large eddy simulation: PTerodaC3TILES version 1.0
Positive matrix factorization of large real-time atmospheric mass spectrometry datasets using error-weighted randomized hierarchical alternating least squares
Numerical simulations of ocean surface waves along the Australian coast with a focus on the Great Barrier Reef
CLAQC v1.0 – Country Level Air Quality Calculator: an empirical modeling approach
Hydro-geomorphological modelling of leaky wooden dam efficacy from reach to catchment scale with CAESAR-Lisflood 1.9j
Stabilized two-phase material point method for hydromechanical coupling problems in solid-fluid porous media
Enhancing single precision with quasi-double precision: achieving double-precision accuracy in the Model for Prediction Across Scales – Atmosphere (MPAS-A) version 8.2.1
Advances in land surface forecasting: a comparison of LSTM, gradient boosting, and feed-forward neural networks as prognostic state emulators in a case study with ecLand
Subgrid corrections for the linear inertial equations of a compound flood model – a case study using SFINCS 2.1.1 Dollerup release
Introducing Iterative Model Calibration (IMC) v1.0: a generalizable framework for numerical model calibration with a CAESAR-Lisflood case study
Development of a high-order global dynamical core using the discontinuous Galerkin method for an atmospheric large-eddy simulation (LES) and proposal of test cases: SCALE-DG v0.8.0
An extension of the WeatherBench 2 to binary hydroclimatic forecasts
A joint reconstruction and model selection approach for large-scale linear inverse modeling (msHyBR v2)
Assimilation of snow water equivalent from AMSR2 and IMS satellite data utilizing the local ensemble transform Kalman filter
The Paleochrono-1.1 probabilistic model to derive a common age model for several paleoclimatic sites using absolute and relative dating constraints
Models of buoyancy-driven dykes using continuum plasticity and fracture mechanics: a comparison
Explicit stochastic advection algorithms for the regional-scale particle-resolved atmospheric aerosol model WRF-PartMC (v1.0)
GraphFlow v1.0: approximating groundwater contaminant transport with graph-based methods – an application to fault scenario selection
The Measurement Error Proxy System Model: MEPSM v0.2
Numerical stabilization methods for level-set-based ice front migration
Modelling chemical advection during magma ascent
Consistent point data assimilation in Firedrake and Icepack
A computationally efficient parameterization of aerosol, cloud and precipitation pH for application at global and regional scale (EQSAM4Clim-v12)
Assessing the benefits of approximately exact step sizes for Picard and Newton solver in simulating ice flow (FEniCS-full-Stokes v.1.3.2)
Assessing effects of climate and technology uncertainties in large natural resource allocation problems
Isogeometric analysis of the lithosphere under topographic loading: Igalith v1.0.0
VISIR-2: ship weather routing in Python
Incremental analysis update (IAU) in the Model for Prediction Across Scales coupled with the Joint Effort for Data assimilation Integration (MPAS–JEDI 2.0.0)
Decision-making strategies implemented in SolFinder 1.0 to identify eco-efficient aircraft trajectories: application study in AirTraf 3.0
Developing meshing workflows in Gmsh v4.11 for the geologic uncertainty assessment of high-temperature aquifer thermal energy storage
Development and preliminary validation of a land surface image assimilation system based on the Common Land Model
NorSand4AI: a comprehensive triaxial test simulation database for NorSand constitutive model materials
ParticleDA.jl v.1.0: a distributed particle-filtering data assimilation package
HETerogeneous vectorized or Parallel (HETPv1.0): an updated inorganic heterogeneous chemistry solver for the metastable-state NH4+–Na+–Ca2+–K+–Mg2+–SO42−–NO3−–Cl−–H2O system based on ISORROPIA II
Three-dimensional geological modelling of igneous intrusions in LoopStructural v1.5.10
Estimating volcanic ash emissions using retrieved satellite ash columns and inverse ash transport modeling using VolcanicAshInversion v1.2.1, within the operational eEMEP (emergency European Monitoring and Evaluation Programme) volcanic plume forecasting system (version rv4_17)
Accounting for uncertainties in forecasting tropical-cyclone-induced compound flooding
An automatic mesh generator for coupled 1D–2D hydrodynamic models
Numerical coupling of aerosol emissions, dry removal, and turbulent mixing in the E3SM Atmosphere Model version 1 (EAMv1) – Part 1: Dust budget analyses and the impacts of a revised coupling scheme
Numerical coupling of aerosol emissions, dry removal, and turbulent mixing in the E3SM Atmosphere Model version 1 (EAMv1) – Part 2: A semi-discrete error analysis framework for assessing coupling schemes
jsmetrics v0.2.0: a Python package for metrics and algorithms used to identify or characterise atmospheric jet streams
P3D-BRNS v1.0.0: a three-dimensional, multiphase, multicomponent, pore-scale reactive transport modelling package for simulating biogeochemical processes in subsurface environments
MinVoellmy v1: a lightweight model for simulating rapid mass movements based on a modified Voellmy rheology
Scalable Feature Extraction and Tracking (SCAFET): a general framework for feature extraction from large climate data sets
Sweep interpolation: a cost-effective semi-Lagrangian scheme in the Global Environmental Multiscale model
CHONK 1.0: landscape evolution framework: cellular automata meets graph theory
Perspectives of physics-based machine learning strategies for geoscientific applications governed by partial differential equations
Calibration of absorbing boundary layers for geoacoustic wave modeling in pseudo-spectral time-domain methods
Joshua Hope-Collins, Abdalaziz Hamdan, Werner Bauer, Lawrence Mitchell, and Colin Cotter
Geosci. Model Dev., 18, 4535–4569, https://doi.org/10.5194/gmd-18-4535-2025, https://doi.org/10.5194/gmd-18-4535-2025, 2025
Short summary
Short summary
Effectively using modern supercomputers requires massively parallel algorithms. Time-parallel algorithms calculate the system state (e.g. the atmosphere) at multiple times simultaneously and have exciting potential but are tricky to implement and still require development. We have developed software to simplify implementing and testing the ParaDiag algorithm on supercomputers. We show that for some atmospheric problems it can enable faster or more accurate solutions than traditional techniques.
Raphael Dreger, Timo Kirfel, Andrea Pozzer, Simon Rosanka, Rolf Sander, and Domenico Taraborrelli
Geosci. Model Dev., 18, 4273–4291, https://doi.org/10.5194/gmd-18-4273-2025, https://doi.org/10.5194/gmd-18-4273-2025, 2025
Short summary
Short summary
Model simulations are essential for understanding the interactions between atmospheric composition and weather. However, models including chemistry are very slow. Hence, any computation speedup of such models is important for understanding the role of atmospheric chemistry within the Earth system. In this study we analyzed and optimized the time step for chemistry calculations. Our results show that atmospheric models could be run notably faster without any loss in accuracy.
John Thuburn
Geosci. Model Dev., 18, 3331–3357, https://doi.org/10.5194/gmd-18-3331-2025, https://doi.org/10.5194/gmd-18-3331-2025, 2025
Short summary
Short summary
A new computational fluid dynamics code for simulating the atmospheric boundary layer and convection is presented. Moist thermodynamics is formulated via thermodynamic potentials, avoiding inconsistencies that can be introduced with conventional approaches. Numerical methods typical of weather and climate models are used, with no explicit subgrid scheme. Results highlight some advantages (e.g. large time steps) and disadvantages (e.g. weak vertical fluxes near the surface) of this approach.
Benjamin C. Sapper, Sean Youn, Daven K. Henze, Manjula Canagaratna, Harald Stark, and Jose L. Jimenez
Geosci. Model Dev., 18, 2891–2919, https://doi.org/10.5194/gmd-18-2891-2025, https://doi.org/10.5194/gmd-18-2891-2025, 2025
Short summary
Short summary
Positive matrix factorization (PMF) has been used by atmospheric scientists to extract underlying factors present in large datasets. This paper presents a new technique for error-weighted PMF that drastically reduces the computational costs of previously developed algorithms. We use this technique to deliver interpretable factors and solution diagnostics from an atmospheric chemistry dataset.
Xianghui Dong, Qingxiang Liu, Stefan Zieger, Alberto Alberello, Ali Abdolali, Jian Sun, Kejian Wu, and Alexander V. Babanin
EGUsphere, https://doi.org/10.5194/egusphere-2025-698, https://doi.org/10.5194/egusphere-2025-698, 2025
Short summary
Short summary
Ocean surface wave research is vital for coastal management, marine ecology, and ocean engineering. This study simulates waves along the Australian coast using advanced physical and numerical schemes. Model verification with altimeter and buoy data shows good performance. A two-step parameterization improves accuracy in the complex Great Barrier Reef. This study will help us better understand coastal wave climates and assess sea states, enabling us to better develop, protect, and use the sea.
Stefania Renna, Francesco Granella, Lara Aleluia Reis, and Paulina Schulz-Antipa
Geosci. Model Dev., 18, 2373–2408, https://doi.org/10.5194/gmd-18-2373-2025, https://doi.org/10.5194/gmd-18-2373-2025, 2025
Short summary
Short summary
The Country Level Air Quality Calculator (CLAQC) is a new fast modeling tool that predicts globally country-level monthly and annual concentrations of two major air pollutants, fine particulate matter (PM2.5) and tropospheric ozone (O3). It was designed to inform national and regional climate and pollution mitigation policies. It is easy to use and computationally efficient, allowing for the simulation of a large number of emission scenarios for policy assessments and optimization frameworks.
Joshua M. Wolstenholme, Christopher J. Skinner, David Milan, Robert E. Thomas, and Daniel R. Parsons
Geosci. Model Dev., 18, 1395–1411, https://doi.org/10.5194/gmd-18-1395-2025, https://doi.org/10.5194/gmd-18-1395-2025, 2025
Short summary
Short summary
Leaky wooden dams are a type of natural flood management intervention that aims to reduce flood risk downstream by temporarily holding back water during a storm event and releasing it afterwards. These structures alter the river hydrology, and therefore the geomorphology, yet often this is excluded from numerical models. Here we show that by not simulating geomorphology, we are currently underestimating the efficacy of these structures to reduce the flood peak and store water.
Xiong Tang, Wei Liu, Siming He, Lei Zhu, Michel Jaboyedoff, Huanhuan Zhang, Yuqing Sun, and Zenan Huo
EGUsphere, https://doi.org/10.5194/egusphere-2025-707, https://doi.org/10.5194/egusphere-2025-707, 2025
Short summary
Short summary
This manuscript presents a numerical model about an explicit stabilized coupled two phase material point method. The novelty lies in the employment of stabilized techniques including the strain smoothing method and the multi-field variational principleto to mitigate the spurious pore pressure and maintain the numerical stability. The proposed formulation provides an effective and reliable approach for simulating solid-fluid porous media under static and dynamic conditions.
Jiayi Lai, Lanning Wang, Qizhong Wu, Yizhou Yang, and Fang Wang
Geosci. Model Dev., 18, 1089–1102, https://doi.org/10.5194/gmd-18-1089-2025, https://doi.org/10.5194/gmd-18-1089-2025, 2025
Short summary
Short summary
High-performance computing limitations often hinder numerical model development. Traditional models use double precision for accuracy, which is computationally expensive. Lower precision reduces costs but can introduce errors. The quasi-double-precision (QDP) algorithm helps mitigate these errors. This study applies the QDP algorithm to the Model for Prediction Across Scales – Atmosphere, showing reduced errors and computational time, making it an efficient solution for large-scale simulations.
Marieke Wesselkamp, Matthew Chantry, Ewan Pinnington, Margarita Choulga, Souhail Boussetta, Maria Kalweit, Joschka Bödecker, Carsten F. Dormann, Florian Pappenberger, and Gianpaolo Balsamo
Geosci. Model Dev., 18, 921–937, https://doi.org/10.5194/gmd-18-921-2025, https://doi.org/10.5194/gmd-18-921-2025, 2025
Short summary
Short summary
We compared spatiotemporal forecasts of three machine learning models that learned water and energy
states on the land surface from a physical model scheme. The forecasting models were developed with reanalysis data and simulations on a European scale and transferred to the globe. We found that all approaches deliver highly accurate approximations of the physical dynamic at long time horizons, implying their usefulness to advance land surface forecasting with synthetic data.
states on the land surface from a physical model scheme. The forecasting models were developed with reanalysis data and simulations on a European scale and transferred to the globe. We found that all approaches deliver highly accurate approximations of the physical dynamic at long time horizons, implying their usefulness to advance land surface forecasting with synthetic data.
Maarten van Ormondt, Tim Leijnse, Roel de Goede, Kees Nederhoff, and Ap van Dongeren
Geosci. Model Dev., 18, 843–861, https://doi.org/10.5194/gmd-18-843-2025, https://doi.org/10.5194/gmd-18-843-2025, 2025
Short summary
Short summary
Accurate flood risk assessments are crucial for storm protection. To achieve efficiency, computational costs must be minimized. This paper introduces a novel subgrid approach for linear inertial equations (LIEs) with bed level and friction variations, implemented in the Super-Fast INundation of CoastS (SFINCS) model. Pre-processed lookup tables enhance simulation precision with lower costs. Validations show significant accuracy improvement even at coarser resolutions.
Chayan Banerjee, Kien Nguyen, Clinton Fookes, Gregory Hancock, and Thomas Coulthard
Geosci. Model Dev., 18, 803–818, https://doi.org/10.5194/gmd-18-803-2025, https://doi.org/10.5194/gmd-18-803-2025, 2025
Short summary
Short summary
In geosciences, the reliance on numerical models necessitates the precise calibration of their parameters to effectively translate information from observed to unobserved settings. We introduce a generalizable framework for calibrating numerical models, with a case study of the geomorphological model CAESAR-Lisflood. This approach efficiently identifies the optimal set of parameters for a given numerical model, enabling retrospective and prospective analyses at various temporal resolutions.
Yuta Kawai and Hirofumi Tomita
Geosci. Model Dev., 18, 725–762, https://doi.org/10.5194/gmd-18-725-2025, https://doi.org/10.5194/gmd-18-725-2025, 2025
Short summary
Short summary
When considering future high-resolution atmospheric simulations with O(10–100 m) grid spacing, such as global large-eddy simulations, numerical errors with the conventional low-order dynamical cores can contaminate the effects of turbulent parameterization. To achieve high-order discretization, we developed a global atmospheric dynamical core using the discontinuous Galerkin method (DGM). By conducting several validation tests, we discussed the impact of a high-order DGM on atmospheric flows.
Tongtiegang Zhao, Qiang Li, Tongbi Tu, and Xiaohong Chen
EGUsphere, https://doi.org/10.5194/egusphere-2025-3, https://doi.org/10.5194/egusphere-2025-3, 2025
Short summary
Short summary
The recent WeatherBench 2 provides a versatile framework for the verification of deterministic and ensemble forecasts. In this paper, we present an explicit extension to binary forecasts of hydroclimatic extremes. Sixteen verification metrics for binary forecasts are employed and scorecards are generated to showcase the predictive performance. The extension facilitates more comprehensive comparisons of hydroclimatic forecasts and provides useful information for forecast applications.
Malena Sabaté Landman, Julianne Chung, Jiahua Jiang, Scot M. Miller, and Arvind K. Saibaba
Geosci. Model Dev., 17, 8853–8872, https://doi.org/10.5194/gmd-17-8853-2024, https://doi.org/10.5194/gmd-17-8853-2024, 2024
Short summary
Short summary
Making an informed decision about what prior information to incorporate or discard in an inverse model is important yet very challenging, as it is often not straightforward to distinguish between informative and non-informative variables. In this study, we develop a new approach for incorporating prior information in an inverse model using predictor variables, while simultaneously selecting the relevant predictor variables for the estimation of the unknown quantity of interest.
Joonlee Lee, Myong-In Lee, Sunlae Tak, Eunkyo Seo, and Yong-Keun Lee
Geosci. Model Dev., 17, 8799–8816, https://doi.org/10.5194/gmd-17-8799-2024, https://doi.org/10.5194/gmd-17-8799-2024, 2024
Short summary
Short summary
We developed an advanced snow water equivalent (SWE) data assimilation framework using satellite data based on a land surface model. The results of this study highlight the beneficial impact of data assimilation by effectively combining land surface model and satellite-derived data according to their relative uncertainty, thereby controlling not only transitional regions but also the regions with heavy snow accumulation that are difficult to detect by satellite.
Frédéric Parrenin, Marie Bouchet, Christo Buizert, Emilie Capron, Ellen Corrick, Russell Drysdale, Kenji Kawamura, Amaëlle Landais, Robert Mulvaney, Ikumi Oyabu, and Sune Olander Rasmussen
Geosci. Model Dev., 17, 8735–8750, https://doi.org/10.5194/gmd-17-8735-2024, https://doi.org/10.5194/gmd-17-8735-2024, 2024
Short summary
Short summary
The Paleochrono-1.1 probabilistic dating model allows users to derive a common and optimized chronology for several paleoclimatic sites from various archives (ice cores, speleothems, marine cores, lake cores, etc.). It combines prior sedimentation scenarios with chronological information such as dated horizons, dated intervals, stratigraphic links and (for ice cores) Δdepth observations. Paleochrono-1.1 is available under an open-source license.
Yuan Li, Timothy Davis, Adina E. Pusok, and Richard F. Katz
EGUsphere, https://doi.org/10.5194/egusphere-2024-3504, https://doi.org/10.5194/egusphere-2024-3504, 2024
Short summary
Short summary
Magmatic dykes transport magma to the Earth's surface, sometimes causing eruptions. We advanced a model of dyking, treating it as plastic deformation in a porous medium, unlike the classic model that treats dykes as fractures in elastic solids. Comparing the two, we found the plastic model aligns with the fracture model in dyke speed and energy consumption, despite quantitative differences. This new method could be a powerful tool for understanding volcanic processes during tectonic activity.
Jeffrey H. Curtis, Nicole Riemer, and Matthew West
Geosci. Model Dev., 17, 8399–8420, https://doi.org/10.5194/gmd-17-8399-2024, https://doi.org/10.5194/gmd-17-8399-2024, 2024
Short summary
Short summary
This paper introduces a numerical method for simulating particle-based aerosol transport in atmospheric models. We detail the various numerical properties of the advection order method and demonstrate its implementation in a 3D weather prediction model (WRF) for the first time. Particle-based techniques improve the accuracy of aerosol size and composition predictions, which are key for aerosol–cloud and aerosol–radiation interactions.
Léonard Moracchini, Guillaume Pirot, Kerry Bardot, Mark W. Jessell, and James L. McCallum
Geosci. Model Dev. Discuss., https://doi.org/10.5194/gmd-2024-154, https://doi.org/10.5194/gmd-2024-154, 2024
Revised manuscript accepted for GMD
Short summary
Short summary
To facilitate the exploration of alternative hydrogeological scenarios, we propose to approximate costly physical simulations of contaminant transport by more affordable shortest distances computations. It enables to accept or reject scenarios within a predefined confidence interval. In particular, it can allow to estimate the probability of a fault acting as a preferential path or a barrier.
Matt J. Fischer
Geosci. Model Dev., 17, 6745–6760, https://doi.org/10.5194/gmd-17-6745-2024, https://doi.org/10.5194/gmd-17-6745-2024, 2024
Short summary
Short summary
In paleoclimate research, proxy system models (PSMs) model chemical or biological systems which receive environmental input and output (e.g., geochemical signals). The environmental inputs are rarely noiseless, which causes problems when calibrating multi-input PSMs. Here a PSM is developed which includes generalized noise in both model inputs and outputs and prior information. A quasi-Bayesian method enhances the stability of the solution of the Measurement Error Proxy System Model.
Gong Cheng, Mathieu Morlighem, and G. Hilmar Gudmundsson
Geosci. Model Dev., 17, 6227–6247, https://doi.org/10.5194/gmd-17-6227-2024, https://doi.org/10.5194/gmd-17-6227-2024, 2024
Short summary
Short summary
We conducted a comprehensive analysis of the stabilization and reinitialization techniques currently employed in ISSM and Úa for solving level-set equations, specifically those related to the dynamic representation of moving ice fronts within numerical ice sheet models. Our results demonstrate that the streamline upwind Petrov–Galerkin (SUPG) method outperforms the other approaches. We found that excessively frequent reinitialization can lead to exceptionally high errors in simulations.
Hugo Dominguez, Nicolas Riel, and Pierre Lanari
Geosci. Model Dev., 17, 6105–6122, https://doi.org/10.5194/gmd-17-6105-2024, https://doi.org/10.5194/gmd-17-6105-2024, 2024
Short summary
Short summary
Predicting the behaviour of magmatic systems is important for understanding Earth's matter and heat transport. Numerical modelling is a technique that can predict complex systems at different scales of space and time by solving equations using various techniques. This study tests four algorithms to find the best way to transport the melt composition. The "weighted essentially non-oscillatory" algorithm emerges as the best choice, minimising errors and preserving system mass well.
Reuben W. Nixon-Hill, Daniel Shapero, Colin J. Cotter, and David A. Ham
Geosci. Model Dev., 17, 5369–5386, https://doi.org/10.5194/gmd-17-5369-2024, https://doi.org/10.5194/gmd-17-5369-2024, 2024
Short summary
Short summary
Scientists often use models to study complex processes, like the movement of ice sheets, and compare them to measurements for estimating quantities that are hard to measure. We highlight an approach that ensures accurate results from point data sources (e.g. height measurements) by evaluating the numerical solution at true point locations. This method improves accuracy, aids communication between scientists, and is well-suited for integration with specialised software that automates processes.
Swen Metzger, Samuel Rémy, Jason E. Williams, Vincent Huijnen, and Johannes Flemming
Geosci. Model Dev., 17, 5009–5021, https://doi.org/10.5194/gmd-17-5009-2024, https://doi.org/10.5194/gmd-17-5009-2024, 2024
Short summary
Short summary
EQSAM4Clim has recently been revised to provide an accurate and efficient method for calculating the acidity of atmospheric particles. It is based on an analytical concept that is sufficiently fast and free of numerical noise, which makes it attractive for air quality forecasting. Version 12 allows the calculation of aerosol composition based on the gas–liquid–solid and the reduced gas–liquid partitioning with the associated water uptake for both cases, including the acidity of the aerosols.
Niko Schmidt, Angelika Humbert, and Thomas Slawig
Geosci. Model Dev., 17, 4943–4959, https://doi.org/10.5194/gmd-17-4943-2024, https://doi.org/10.5194/gmd-17-4943-2024, 2024
Short summary
Short summary
Future sea-level rise is of big significance for coastal regions. The melting and acceleration of glaciers plays a major role in sea-level change. Computer simulation of glaciers costs a lot of computational resources. In this publication, we test a new way of simulating glaciers. This approach produces the same results but has the advantage that it needs much less computation time. As simulations can be obtained with fewer computation resources, higher resolution and physics become affordable.
Jevgenijs Steinbuks, Yongyang Cai, Jonas Jaegermeyr, and Thomas W. Hertel
Geosci. Model Dev., 17, 4791–4819, https://doi.org/10.5194/gmd-17-4791-2024, https://doi.org/10.5194/gmd-17-4791-2024, 2024
Short summary
Short summary
This paper applies a cutting-edge numerical method, SCEQ, to show how uncertain climate change and technological progress affect the future utilization of the world's scarce land resources. The paper's key insight is to illustrate how much global cropland will expand when future crop yields are unknown. The study finds the range of outcomes for land use change to be smaller when using this novel method compared to existing deterministic models.
Rozan Rosandi, Yudi Rosandi, and Bernd Simeon
EGUsphere, https://doi.org/10.5194/egusphere-2024-1093, https://doi.org/10.5194/egusphere-2024-1093, 2024
Short summary
Short summary
We model Earth's lithosphere as a thin elastic shell and present numerical methods of isogeometric finite element analysis to simulate its deformation in isostatic equilibrium using technologies from computer-aided design. The simulations also serve as a basis for identifying parameters of the model that are most plausible to explain observed data. This research has been done to showcase the capabilities of isogeometric analysis in solving higher-order problems in geoscientific applications.
Gianandrea Mannarini, Mario Leonardo Salinas, Lorenzo Carelli, Nicola Petacco, and Josip Orović
Geosci. Model Dev., 17, 4355–4382, https://doi.org/10.5194/gmd-17-4355-2024, https://doi.org/10.5194/gmd-17-4355-2024, 2024
Short summary
Short summary
Ship weather routing has the potential to reduce CO2 emissions, but it currently lacks open and verifiable research. The Python-refactored VISIR-2 model considers currents, waves, and wind to optimise routes. The model was validated, and its computational performance is quasi-linear. For a ferry sailing in the Mediterranean Sea, VISIR-2 yields the largest percentage emission savings for upwind navigation. Given the vessel performance curve, the model is generalisable across various vessel types.
Soyoung Ha, Jonathan J. Guerrette, Ivette Hernández Baños, William C. Skamarock, and Michael G. Duda
Geosci. Model Dev., 17, 4199–4211, https://doi.org/10.5194/gmd-17-4199-2024, https://doi.org/10.5194/gmd-17-4199-2024, 2024
Short summary
Short summary
To mitigate the imbalances in the initial conditions, this study introduces our recent implementation of the incremental analysis update (IAU) in the Model for Prediction Across Scales – Atmospheric (MPAS-A) component coupled with the Joint Effort for Data assimilation Integration (JEDI) through the cycling system. A month-long cycling run demonstrates the successful implementation of the IAU capability in the MPAS–JEDI cycling system.
Federica Castino, Feijia Yin, Volker Grewe, Hiroshi Yamashita, Sigrun Matthes, Simone Dietmüller, Sabine Baumann, Manuel Soler, Abolfazl Simorgh, Maximilian Mendiguchia Meuser, Florian Linke, and Benjamin Lührs
Geosci. Model Dev., 17, 4031–4052, https://doi.org/10.5194/gmd-17-4031-2024, https://doi.org/10.5194/gmd-17-4031-2024, 2024
Short summary
Short summary
We introduce SolFinder 1.0, a decision-making tool to select trade-offs between different objective functions for optimal aircraft trajectories, including fuel use, flight time, NOx emissions, contrail distance, and climate impact. The module is included in the AirTraf 3.0 submodel and uses weather conditions simulated by the EMAC atmospheric model. This paper focuses on the ability of SolFinder to identify eco-efficient trajectories, reducing a flight's climate impact at limited cost penalties.
Ali Dashti, Jens C. Grimmer, Christophe Geuzaine, Florian Bauer, and Thomas Kohl
Geosci. Model Dev., 17, 3467–3485, https://doi.org/10.5194/gmd-17-3467-2024, https://doi.org/10.5194/gmd-17-3467-2024, 2024
Short summary
Short summary
This study developed new meshing workflows to enable the automatic generation of meshes that follow geological models. The workflow allows for importing several geological models as input for Gmsh and later exporting the same number of high-quality meshes. This way, geological uncertainty is directly included in the numerical simulations. This study evaluates the impact of the geological uncertainty on thermohydraulic performance of two reservoirs for high-temperature heat storage applications.
Wangbin Shen, Zhaohui Lin, Zhengkun Qin, and Juan Li
Geosci. Model Dev., 17, 3447–3465, https://doi.org/10.5194/gmd-17-3447-2024, https://doi.org/10.5194/gmd-17-3447-2024, 2024
Short summary
Short summary
In this study, a land surface image assimilation system capable of optimizing the spatial structure of the background field is constructed by introducing the curvelet analysis method and taking the similarity of image structure as a weak constraint. The findings demonstrate that the assimilation of surface soil moisture observation images effectively and reasonably enhances the spatial structure of soil moisture analysis field.
Luan Carlos de Sena Monteiro Ozelim, Michéle Dal Toé Casagrande, and André Luís Brasil Cavalcante
Geosci. Model Dev., 17, 3175–3197, https://doi.org/10.5194/gmd-17-3175-2024, https://doi.org/10.5194/gmd-17-3175-2024, 2024
Short summary
Short summary
The paper addresses synthetic dataset challenges in nonlinear constitutive modeling of soils, providing two datasets: one with 2000 soil types, 40 test conditions each (160 000 triaxial tests), and a second with 2048 soil types, 42 test conditions each (172 032 triaxial tests). Each dataset is a 4000×10 matrix applicable for multivariate forecasting and geotechnical simulations. In addition, a new Python code is introduced, empowering researchers to leverage Python packages for NorSand analyses.
Daniel Giles, Matthew M. Graham, Mosè Giordano, Tuomas Koskela, Alexandros Beskos, and Serge Guillas
Geosci. Model Dev., 17, 2427–2445, https://doi.org/10.5194/gmd-17-2427-2024, https://doi.org/10.5194/gmd-17-2427-2024, 2024
Short summary
Short summary
Digital twins of physical and human systems informed by real-time data are becoming ubiquitous across a wide range of settings. Progress for researchers is currently limited by a lack of tools to run these models effectively and efficiently. A key challenge is the optimal use of high-performance computing environments. The work presented here focuses on a developed open-source software platform which aims to improve this usage, with an emphasis placed on flexibility, efficiency, and scalability.
Stefan J. Miller, Paul A. Makar, and Colin J. Lee
Geosci. Model Dev., 17, 2197–2219, https://doi.org/10.5194/gmd-17-2197-2024, https://doi.org/10.5194/gmd-17-2197-2024, 2024
Short summary
Short summary
This work outlines a new solver written in Fortran to calculate the partitioning of metastable aerosols at thermodynamic equilibrium based on the forward algorithms of ISORROPIA II. The new code includes numerical improvements that decrease the computational speed (compared to ISORROPIA II) while improving the accuracy of the partitioning solution.
Fernanda Alvarado-Neves, Laurent Ailleres, Lachlan Grose, Alexander R. Cruden, and Robin Armit
Geosci. Model Dev., 17, 1975–1993, https://doi.org/10.5194/gmd-17-1975-2024, https://doi.org/10.5194/gmd-17-1975-2024, 2024
Short summary
Short summary
Previous work has demonstrated that adding geological knowledge to modelling methods creates more accurate and reliable models. Following this reasoning, we added constraints from magma emplacement mechanisms into existing modelling frameworks to improve the 3D characterisation of igneous intrusions. We tested the method on synthetic and real-world case studies, and the results show that our method can reproduce intrusion morphologies with no manual processing and using realistic datasets.
André R. Brodtkorb, Anna Benedictow, Heiko Klein, Arve Kylling, Agnes Nyiri, Alvaro Valdebenito, Espen Sollum, and Nina Kristiansen
Geosci. Model Dev., 17, 1957–1974, https://doi.org/10.5194/gmd-17-1957-2024, https://doi.org/10.5194/gmd-17-1957-2024, 2024
Short summary
Short summary
It is vital to know the extent and concentration of volcanic ash in the atmosphere during a volcanic eruption. Whilst satellite imagery may give an estimate of the ash right now (assuming no cloud coverage), we also need to know where it will be in the coming hours. This paper presents a method for estimating parameters for a volcanic eruption based on satellite observations of ash in the atmosphere. The software package is open source and applicable to similar inversion scenarios.
Kees Nederhoff, Maarten van Ormondt, Jay Veeramony, Ap van Dongeren, José Antonio Álvarez Antolínez, Tim Leijnse, and Dano Roelvink
Geosci. Model Dev., 17, 1789–1811, https://doi.org/10.5194/gmd-17-1789-2024, https://doi.org/10.5194/gmd-17-1789-2024, 2024
Short summary
Short summary
Forecasting tropical cyclones and their flooding impact is challenging. Our research introduces the Tropical Cyclone Forecasting Framework (TC-FF), enhancing cyclone predictions despite uncertainties. TC-FF generates global wind and flood scenarios, valuable even in data-limited regions. Applied to cases like Cyclone Idai, it showcases potential in bettering disaster preparation, marking progress in handling cyclone threats.
Younghun Kang and Ethan J. Kubatko
Geosci. Model Dev., 17, 1603–1625, https://doi.org/10.5194/gmd-17-1603-2024, https://doi.org/10.5194/gmd-17-1603-2024, 2024
Short summary
Short summary
Models used to simulate the flow of coastal and riverine waters, including flooding, require a geometric representation (or mesh) of geographic features that exhibit a range of disparate spatial scales from large, open waters to small, narrow channels. Representing these features in an accurate way without excessive computational overhead presents a challenge. Here, we develop an automatic mesh generation tool to help address this challenge. Our results demonstrate the efficacy of our approach.
Hui Wan, Kai Zhang, Christopher J. Vogl, Carol S. Woodward, Richard C. Easter, Philip J. Rasch, Yan Feng, and Hailong Wang
Geosci. Model Dev., 17, 1387–1407, https://doi.org/10.5194/gmd-17-1387-2024, https://doi.org/10.5194/gmd-17-1387-2024, 2024
Short summary
Short summary
Sophisticated numerical models of the Earth's atmosphere include representations of many physical and chemical processes. In numerical simulations, these processes need to be calculated in a certain sequence. This study reveals the weaknesses of the sequence of calculations used for aerosol processes in a global atmosphere model. A revision of the sequence is proposed and its impacts on the simulated global aerosol climatology are evaluated.
Christopher J. Vogl, Hui Wan, Carol S. Woodward, and Quan M. Bui
Geosci. Model Dev., 17, 1409–1428, https://doi.org/10.5194/gmd-17-1409-2024, https://doi.org/10.5194/gmd-17-1409-2024, 2024
Short summary
Short summary
Generally speaking, accurate climate simulation requires an accurate evolution of the underlying mathematical equations on large computers. The equations are typically formulated and evolved in process groups. Process coupling refers to how the evolution of each group is combined with that of other groups to evolve the full set of equations for the whole atmosphere. This work presents a mathematical framework to evaluate methods without the need to first implement the methods.
Tom Keel, Chris Brierley, and Tamsin Edwards
Geosci. Model Dev., 17, 1229–1247, https://doi.org/10.5194/gmd-17-1229-2024, https://doi.org/10.5194/gmd-17-1229-2024, 2024
Short summary
Short summary
Jet streams are an important control on surface weather as their speed and shape can modify the properties of weather systems. Establishing trends in the operation of jet streams may provide some indication of the future of weather in a warming world. Despite this, it has not been easy to establish trends, as many methods have been used to characterise them in data. We introduce a tool containing various implementations of jet stream statistics and algorithms that works in a standardised manner.
Amir Golparvar, Matthias Kästner, and Martin Thullner
Geosci. Model Dev., 17, 881–898, https://doi.org/10.5194/gmd-17-881-2024, https://doi.org/10.5194/gmd-17-881-2024, 2024
Short summary
Short summary
Coupled reaction transport modelling is an established and beneficial method for studying natural and synthetic porous material, with applications ranging from industrial processes to natural decompositions in terrestrial environments. Up to now, a framework that explicitly considers the porous structure (e.g. from µ-CT images) for modelling the transport of reactive species is missing. We presented a model that overcomes this limitation and represents a novel numerical simulation toolbox.
Stefan Hergarten
Geosci. Model Dev., 17, 781–794, https://doi.org/10.5194/gmd-17-781-2024, https://doi.org/10.5194/gmd-17-781-2024, 2024
Short summary
Short summary
The Voellmy rheology has been widely used for simulating snow and rock avalanches. Recently, a modified version of this rheology was proposed, which turned out to be able to predict the observed long runout of large rock avalanches theoretically. The software MinVoellmy presented here is the first numerical implementation of the modified rheology. It consists of MATLAB and Python classes, where simplicity and parsimony were the design goals.
Arjun Babu Nellikkattil, Danielle Lemmon, Travis Allen O'Brien, June-Yi Lee, and Jung-Eun Chu
Geosci. Model Dev., 17, 301–320, https://doi.org/10.5194/gmd-17-301-2024, https://doi.org/10.5194/gmd-17-301-2024, 2024
Short summary
Short summary
This study introduces a new computational framework called Scalable Feature Extraction and Tracking (SCAFET), designed to extract and track features in climate data. SCAFET stands out by using innovative shape-based metrics to identify features without relying on preconceived assumptions about the climate model or mean state. This approach allows more accurate comparisons between different models and scenarios.
Mohammad Mortezazadeh, Jean-François Cossette, Ashu Dastoor, Jean de Grandpré, Irena Ivanova, and Abdessamad Qaddouri
Geosci. Model Dev., 17, 335–346, https://doi.org/10.5194/gmd-17-335-2024, https://doi.org/10.5194/gmd-17-335-2024, 2024
Short summary
Short summary
The interpolation process is the most computationally expensive step of the semi-Lagrangian (SL) approach. In this paper we implement a new interpolation scheme into the semi-Lagrangian approach which has the same computational cost as a third-order polynomial scheme but with the accuracy of a fourth-order interpolation scheme. This improvement is achieved by using two third-order backward and forward polynomial interpolation schemes in two consecutive time steps.
Boris Gailleton, Luca C. Malatesta, Guillaume Cordonnier, and Jean Braun
Geosci. Model Dev., 17, 71–90, https://doi.org/10.5194/gmd-17-71-2024, https://doi.org/10.5194/gmd-17-71-2024, 2024
Short summary
Short summary
This contribution presents a new method to numerically explore the evolution of mountain ranges and surrounding areas. The method helps in monitoring with details on the timing and travel path of material eroded from the mountain ranges. It is particularly well suited to studies juxtaposing different domains – lakes or multiple rock types, for example – and enables the combination of different processes.
Denise Degen, Daniel Caviedes Voullième, Susanne Buiter, Harrie-Jan Hendricks Franssen, Harry Vereecken, Ana González-Nicolás, and Florian Wellmann
Geosci. Model Dev., 16, 7375–7409, https://doi.org/10.5194/gmd-16-7375-2023, https://doi.org/10.5194/gmd-16-7375-2023, 2023
Short summary
Short summary
In geosciences, we often use simulations based on physical laws. These simulations can be computationally expensive, which is a problem if simulations must be performed many times (e.g., to add error bounds). We show how a novel machine learning method helps to reduce simulation time. In comparison to other approaches, which typically only look at the output of a simulation, the method considers physical laws in the simulation itself. The method provides reliable results faster than standard.
Carlos Spa, Otilio Rojas, and Josep de la Puente
Geosci. Model Dev., 16, 7237–7252, https://doi.org/10.5194/gmd-16-7237-2023, https://doi.org/10.5194/gmd-16-7237-2023, 2023
Short summary
Short summary
This paper develops a calibration methodology of all absorbing techniques typically used by Fourier pseudo-spectral time-domain (PSTD) methods for geoacoustic wave simulations. The main contributions of the paper are (a) an implementation and quantitative comparison of all absorbing techniques available for PSTD methods through a simple and robust numerical experiment, and (b) a validation of these absorbing techniques in several 3-D seismic scenarios with gradual heterogeneity complexities.
Cited articles
Alnæs, M. S.: UFL: a finite element form language, in: Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, edited by: Logg, A., Mardal, K.-A., and Wells, G. N., chap. 17, Springer, 2012.
Alnæs, M. S., Logg, A., Mardal, K.-A., Skavhaug, O., and Langtangen, H. P.: Unified Framework for Finite Element Assembly, Int. J. Computat. Sci. Eng., 4, 231–244, https://doi.org/10.1504/IJCSE.2009.029160, 2009.
Alnæs, M. S., Logg, A., and Mardal, K.-A.: UFC: a finite element code generation interface, in: Automated Solution of Differential Equations by the Finite Element Method, Volume 84 of Lecture Notes in Computational Science and Engineering, edited by: Logg, A., Mardal, K.-A., and Wells, G. N., chap. 16, Springer, 2012.
Alnæs, M. S., Logg, A., Ølgaard, K. B., Rognes, M. E., and Wells, G. N.: Unified Form Language: a domain-specific language for weak formulations of partial differential equations, ACM Trans. Mathe. Softw., available at: http://arxiv.org/abs/1211.4047 (last access: 13 December 2013), 2013.
Arakawa, A. and Hsu, Y.-J. G.: Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations, Mon. Weather Rev., 118, 1960–1969, 1990.
Arakawa, A. and Lamb, V.: Computational design of the basic dynamical processes of the UCLA general circulation model, Meth. Computat. Phys., 17, 174–267, 1977.
Auricchio, F., Brezzi, F., and Lovadina, C.: Mixed Finite Element Methods, Encyclopedia of Computational Mechanics, 2004.
Barden, D. and Thomas, C.: An Introduction to Differential Manifolds, Imperial College Press, 2003.
Bernard, P.-E., Remacle, J.-F., and Legat, V.: High-order Discontinuous G}alerkin Methods for Solving Conservation Laws on General 2-{D Manifolds, 2008.
Brezzi, F. and Fortin, M.: Mixed and Hybrid Finite Element Methods, vol. 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991.
Brezzi, F., Douglas, J., and Marini, L. D.: Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217–235, 1985.
Cockburn, B. and Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173–261, 2001.
Côté, J.: A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Q. J. Roy. Meteorol. Soc., 114, 1347–1352, 1988.
Cotter, C. and Shipton, J.: Mixed finite elements for numerical weather prediction, J. Computat. Phys., 231, 7076–7091, https://doi.org/10.1016/j.jcp.2012.05.020, 2012.
Cotter, C. J., Ham, D. A., and Pain, C. C.: A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling, Ocean Modell., 26, 86–90, https://doi.org/10.1016/j.ocemod.2008.09.002, 2009.
Dedner, A., Klöfkorn, R., Nolte, M., and Ohlberger, M.: A generic interface for parallel and adaptive discretization schemes: abstraction principles and the DUNE-FEM module, Computing, 90, 165–196, 2010.
DeSimone, A., Heltai, L., and Manigrasso, C.: Tools for the Solution of PDEs Defined on Curved Manifolds with deal.II, International School for Advanced Studies (SISSA), available at: http://hdl.handle.net/1963/3700 (last access: 13 December 2013), 2009.
Giraldo, F. X.: High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere, J. Computat. Phys., 214, 447–465, 2006.
Holm, D.: Geometric Mechanics – Part I: Dynamics and Symmetry, Imperial College Press, 2008.
Karniadakis, G. E. and Sherwin, S. J.: Spectral/hp Element Methods for CFD, Oxford University Press, 1999.
Kirby, R. C.: Algorithm 839: FIAT, a new paradigm for computing finite element basis functions, ACM T. Math. Software, 30, 502–516, https://doi.org/10.1145/1039813.1039820, 2004.
Kirby, R. C. and Logg, A.: A compiler for variational forms, ACM T. Math. Software, 32, 417–444, https://doi.org/10.1145/1163641.1163644, 2006.
Kuptsov, L.: Gram Determinant, in: Encyclopedia of Mathematics, Springer, available at: http://www.encyclopediaofmath.org/index.php/Gram_determinant?%oldid=18442 (last access: 2 July 2013), 2011.
Leimkuhler, B. and Reich, S.: Simulating Hamiltonian Dynamics, chap. 12, CUP, 2005.
Le Roux, D., Sène, A., Rostand, V., and Hanert, E.: On some spurious mode issues in shallow-water models using a linear algebra approach, Ocean Modell., 10, 83–94, 2005.
Logg, A. and Wells, G. N.: DOLFIN: automated finite element computing, ACM T. Math. Software, 37, 20:1–20:28, https://doi.org/10.1145/1731022.1731030, 2010.
Logg, A., Mardal, K.-A., and Wells, G.: Automated Solution of Differential Equations by the Finite Element Method: the Fenics Book, vol. 84, Springer, 2012a.
Logg, A., Mardal, K.-A., and Wells, G. N. (Eds.): Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012b.
Logg, A., Mardal, K.-A., and Wells, G. N.: Finite element assembly, in: Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012c.
Logg, A., Ølgaard, K. B., Rognes, M. E., and Wells, G. N.: FFC: the FEniCS Form Compiler, in: Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, edited by: Logg, A., Mardal, K.-A., and Wells, G. N., chap. 11, Springer, 2012d.
Logg, A., Wells, G. N., and Hake, J.: DOLFIN: a C++/Python Finite Element Library, in: Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, edited by: Logg, A., Mardal, K.-A., and Wells, G. N., chap. 10, Springer, 2012e.
McRae, A. T. T and Cotter, C. J.: Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements, Q. J. R. Meteorol. Soc., http://arxiv.org/abs/1305.4477 (last access: 13 December 2013), 2013.
Nédélec, J.-C.: Mixed finite elements in $\mathbb R^3$, Numer. Math., 35, 315–341, https://doi.org/10.1007/BF01396415, 1980.
Nédélec, J.-C.: A new family of mixed finite elements in $\mathbb R^3$, Numer. Math., 50, 57–81, 1986.
Penrose, R.: A generalized inverse for matrices, Proc. Cambridge Philos. Soc, 51, 406–413, 1955.
Raviart, P.-A. and Thomas, J. M.: A mixed finite element method for 2nd order elliptic problems, in: Mathematical aspects of finite element methods (Proc. C}onf., Consiglio Naz. delle Ricerche (C.{N.R.), Rome, 1975), 292–315, Lecture Notes in Math., vol. 606, Springer, Berlin, 1977.
Richardson, C. N., and Wells, G. N.: Expressive and scalable finite element simulation beyond 1000 cores, HECToR university distributed CSE project report, available at: http://www.hector.ac.uk/cse/distributedcse/reports/UniDOLFIN/, last access: 23 September 2013.
Rognes, M. E., Kirby, R. C., and Logg, A.: Efficient assembly of $H(div)$ and $H(curl)$ conforming finite elements, SIAM J. Sci. Comput., 31, 4130–4151, 2009.
Schmidt, A., Barth, T. J., and Siebert, K. G.: Design of adaptive finite element software: the finite element toolbox ALBERTA, vol. 42, Springer, 2005.
Sherwin, S., Kirby, R. M., and the Nektar++ team, available at: http://www.nektar.info, last access: 2 July 2013.
Williamson, D. L., Drake, J. B., Hack, J. J., Jakob, R., and Swarztrauber, P. N.: A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Computat. Phys., 102, 211–224, 1992.
Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z.: The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, 2005.
