Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model

Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves, while not suffering from severe restrictions on the timestep size. To propagate the state of the atmosphere forward in time, a non-linear equation for the prognostic variables has to be solved at every timestep. Since the nonlinearity is typically weak, this is done with a small number of Newton- or Picard- iterations, which in turn require the efficient solution of a large system on linear equations with O(106 − 109) unknowns. This linear solve is often the computationally most costly part of the model. In this paper an efficient linear solver for the LFRic next-generation model, currently developed by the Met Office, is described. The model uses an advanced mimetic finite element discretisation which makes the construction of efficient solvers challenging compared to models using standard finite-difference and finite-volume methods. The linear solver hinges on a bespoke multigrid preconditioner of the Schur-complement system for the pressure correction. By comparing to Krylov-subspace methods, the superior performance and robustness of the multigrid algorithm is demonstrated for standard test cases and realistic model setups. In production mode, the model will have to run in parallel on 100,000s of processing elements. As confirmed by numerical experiments, one particular advantage of the multigrid solver is its excellent parallel scalability due to avoiding expensive global reduction operations

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

We present a novel second-order semi-implicit hybrid finite volume / finite element (FV/FE) scheme for the numerical solution of the incompressible and weakly compressible Navier-Stokes equations on moving unstructured meshes using an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a suitable splitting of the governing PDE into subsystems and employs staggered grids, where the pressure is defined on the primal simplex mesh, while the velocity and the remaining flow quantities are defined on an edge-based staggered dual mesh. The key idea of the scheme is to discretize the nonlinear convective and viscous terms using an explicit FV scheme that employs the space-time divergence form of the governing equations on moving space-time control volumes. For the convective terms, an ALE extension of the Ducros flux on moving meshes is introduced, which is kinetic energy preserving and stable in the energy norm when adding suitable numerical dissipation terms. Finally, the pressure equation of the Navier-Stokes system is solved on the new mesh configuration using a continuous FE method, with $\mathbb{P}_1$ Lagrange elements. The ALE hybrid FV/FE method is applied to several incompressible test problems ranging from non-hydrostatic free surface flows over a rising bubble to flows over an oscillating cylinder and an oscillating ellipse. Via the simulation of a circular explosion problem on a moving mesh, we show that the scheme applied to the weakly compressible Navier-Stokes equations is able to capture weak shock waves, rarefactions and moving contact discontinuities. We show that our method is particularly efficient for the simulation of weakly compressible flows in the low Mach number limit, compared to a fully explicit ALE schem

Two dimensional computational fluid dynamics model of pollutant transport in an open pit mine under Arctic inversion

Thesis (M.S.) University of Alaska Fairbanks, 2012A better understanding of the microscale meteorology of deep, open pit mines is important for mineral exploitation in arctic and subarctic regions. During strong temperature inversions in the atmospheric boundary layer--which are common in arctic regions during the winter--the concentrations of gaseous pollutants in open pit mines can reach dangerous levels. In this research, a two dimensional computational fluid dynamics (CFD) model was used to study the atmosphere of an open pit mine. The natural airflow patterns in an open pit mine are strongly dependent on the geometry of the mine. Generally, mechanical turbulence created by the mine topography results in a recirculatory region at the bottom of the mine that is detached from the freestream. The presence of a temperature inversion further inhibits natural ventilation in open pit mines, and the air can quickly become contaminated if a source of pollution is present. Several different exhaust fan configurations were modeled to see if the pollution problem could be mitigated. The two dimensional model suggests that mitigation is possible, but the large quantity of ventilating air required would most likely beimpractical in an industrial setting.1. Introduction -- 1.1. Scientific rationale -- 1.2. Air inversion -- 1.3. Previous modeling approaches -- 1.4. Solution approaches -- 1.5. Proposed remediation measures -- 1.6. Scope of this research -- 1.7. Work plan -- 2. Data collection -- 3. Model development -- 3.1. Fundamental transport equations -- 3.2. Cell zone and boundary conditions -- 3.3. Meshing -- 3.4. Discretization -- 3.5. TurbulenceModeling -- 3.6. Geometry and mesh creation -- 3.7. Wind flow in open pit mines -- 3.8. Development of an atmospheric inversion -- 4. Pollutant transport in an open pit mine under Arctic air inversion -- 5. Mitigation of pollutants -- 5.1. Helicopter -- 5.2. Exhaust fan: 142 m³/s -- 5.3. Exhaust fan: 556 m³/s -- 5.4. Exhaust fans: multiple fans, multiple sources (142 m³/s) -- 5.5. Exhaust fans: multiple fans, multiple sources (284 m³/s) -- 6. Summary, conclusions, and recommendations for future work -- 6.1. Summary and conclusions -- 6.2. Future work -- 7. References

Aeronautical engineering: A continuing bibliography with indexes (supplement 216)

This bibliography lists 505 reports, articles and other documents introduced into the NASA scientific and technical information system in July, 1987