SIAM Conference on Mathematical and Computational Issues in the Geosciences
September 11—14, 2017 • Erlangen, Germany
Discontinuous Skeletal Methods for Computational Geosciences
Discontinuous Skeletal methods are based on discrete unknowns that are discontinuous polynomials on the mesh skeleton. Such methods offer several attractive features: a dimension-independent construction, and the use of arbitrary polynomial orders combined with general grids, including non-matching interfaces and polyhedral cell shapes. Positioning unknowns at mesh faces is also a natural way to express fundamental continuum properties at the discrete level such as local mass or force balance. One prominent example of Discontinuous Skeletal methods is the Hybridizable Discontinuous Galerkin method. Recently, another method has emerged : the Hybrid High-Order (HHO) method. HHO methods were originally devised to approximate scalar diffusive and linear elasticity problems, and have undergone a substantial development in terms of analysis and applications. The cornerstone of HHO methods are fully local, reconstruction operators, which can offer reduced computational costs by organizing simulations into (fully parallelizable) local solves and a global transmission problem. In this talk, we aim at providing a (gentle) introduction to the devising and analysis of HHO methods and to briefly touch upon some of the more recent developments.
Bridging Scales in Weather and Climate Models with Adaptive Mesh Refinement Techniques
Jared Ferguson, University of Michigan
Hans Johansen, Peter McCorquodale and Phillip Colella, Lawrence Berkeley National Laboratory
Extreme atmospheric events such as tropical cyclones are inherently complex multi-scale phenomena. Such extremes are a challenge to simulate in conventional atmosphere models which typically use rather coarse uniform-grid resolutions. Adaptive Mesh Refinement (AMR) techniques seek to mitigate these challenges. They dynamically place high-resolution grid patches over user-defined features of interest, thus providing sufficient local resolution over e.g. a developing cyclone while limiting the total computational burden. Studying such techniques in idealized simulations enables the assessment of the AMR approach in a controlled environment and can assist in identifying the effective refinement choices for more complex, realistic simulations.
The talk reviews a newly-developed, non-hydrostatic, finite-volume dynamical core for future-generation weather and climate models. It implements refinement in both space and time on a cubed-sphere grid and is based on the AMR library Chombo, developed by the Lawrence Berkeley National Laboratory. Idealized 2D shallow-water and 3D test cases are discussed including interacting vortices, flows over topography, and a tropical cyclone simulation with simplified moisture processes. These simulations test the effectiveness of both static and dynamic grid refinements as well as the sensitivity of the model results to various adaptation criteria and forcing mechanisms. The AMR results will furthermore be compared to more traditional variable-resolution techniques, such as the use of a statically-nested mesh in NCAR’s Community Atmosphere Model CAM with its Spectral Element (SE) dynamical core. This sheds light on the pros and cons of both approaches.
Methane hydrate modeling, analysis, and simulation: coupled systems and scales
Methane hydrate is an ice-like substance abundantly present in deep ocean sediments and in the Arctic. Geoscientists recognize the tremendous importance of gas hydrate as a crucial element of the global carbon cycle, a contributor to climate change studied in various deep ocean observatories, as well as a possible energy source evaluated in recent pilot engineering projects in the US and Japan. Hydrate evolution however is curiously not very well studied by computational mathematics community.
In the talk we present the challenges of hydrate modeling, which start with the need to respond to the interests of geophysicists to enable lasting collaborations that deliver meaningful results. Next we present a cascade of complex to simplified models. For the latter, some analysis of the underlying well-posedness in a very weak setting can be achieved. For the former, interesting scenarios involving multiple scales, and coupled phenomena of flow, transport, phase transitions, and geomechanics, can be formulated.
I will report on most recent results obtained jointly with the geophysicists Marta Torres (Oregon State), Wei-Li Hong (Arctic University of Norway), mathematicians Ralph Showalter (Oregon State) and F. Patricia Medina (WPI), computational scientist Anna Trykozko (University of Warsaw), as well as many current and former students to be named in the talk.
High resolution atmospheric turbulence simulations for applied problems
Originally applied to study convective atmospheric boundary layers (CBL), large-eddy simulation (LES) is meanwhile used in many fields of science. This is mainly the consequence of a massive increase in available computer resources. State-of-the-art massively parallel computers have opened the field for a wide variety of new applications. On these machines, simulations with extremely large numerical grids of up to 40003 grid points and even more are currently carried out in acceptable time. In Meteorology, beside for the fundamental research of neutral and stable stratified flows, where the typical eddy size is much smaller than for pure convectively driven flows, LES starts to be used also for more applied topics like air pollution modeling, flow around buildings, or wind energy. Moreover, the interaction of turbulence of different scales can be studied for the first time. Lagrangian particle models coupled to LES allow for further interesting applications, e.g. to calculate footprints of turbulence sensors in heterogeneous terrain, or to simulate the effect of turbulence on the growth of cloud droplets. Respective simulations require both, a large model domain size to capture the large scales and a sufficiently fine grid spacing to resolve the interacting smaller scales, creating a very high demand on computational resources.
The talk will start with a short general introduction to LES and will then give an overview of current studies with very high spatial resolution performed at IMUK, like simulations of coherent structures in the convective boundary layer, simulations of the urban environment, and the effect of turbulence on cloud droplet growth or aircraft during takeoff and landing, as well as LES applications for wind energy systems.
Scalable nonlinear and linear solvers for multiphase flow in heterogeneous porous media
Numerical Reservoir Simulation of flow and transport processes in subsurface formations is an integral part of managing underground hydrocarbon and water resources with application areas that include oil/gas recovery, management of water resources, and subsurface CO2 sequestration. We discuss the challenges associated with Reservoir Simulation of multiphase flow in large-scale heterogeneous formations. The focus is on developing scalable (i.e., efficient for heterogeneous, large-scale problems) linear and nonlinear solvers. The discussion is split into two parts: (1) Algebraic Multi-Scale (AMS) linear solvers and (2) trust-region nonlinear solvers. AMS linear solvers are designed to deal with the multiscale distributions of the properties of natural geologic formations (e.g., permeability/conductivity). The complex multiscale spatial variations in the permeability field lead to complex multiscale fluid dynamics (e.g., pressure and velocity). A brief description of two-level AMS formulations for the pressure field is given. Then, we discuss the development of massively parallel AMS linear solvers on multi-core and GPU architectures. We demonstrate that significant progress has been made in our collective ability to capture the impact of multiscale spatial variations in formation properties on the pressure and velocity fields in large subsurface systems.
We then discuss the multiscale nature of the time scales associated with the nonlinear fluid transport processes and the strong nonlinear coupling between the governing conservation equations and constitutive relations. Specifically, we describe a nonlinear solver framework based on constructing trust-regions of the fractional-flow (phase flux) function. We demonstrate that the nonlinearity of the transport (saturation) equations can be resolved efficiently through analysis of the discrete/numerical flux functions. The theory is elaborated for immiscible two-phase fluid flow in highly heterogeneous formations. The robustness and computational efficiency of the flux-based trust-region nonlinear solver are demonstrated using challenging problems in the presence of counter-current flow due to strong buoyancy effects. Combining multiscale linear solvers with trust-region nonlinear solvers has made it possible to simulate multiphase flow and transport processes in problems of growing size (resolution) and complexity. Nevertheless, enormous challenges lie ahead, and we conclude with a perspective on these important and exciting computational geoscience problems.
New frontiers in Earth-System Modelling
The gradual progress in global numerical weather prediction includes a systematic approach to assess and quantify the associated forecast uncertainty by means of high-resolution ensembles of assimilation and forecasts. This involves simulations with billions of gridpoints, the continuous assimilation of billions of observations, rigorous verification, validation and uncertainty quantification, and it involves increasing model complexity through completing the descriptions of the global water and carbon cycles. The research requires a deeper understanding of multi-scale interactions within the atmosphere and oceans, and through interactions at the interfaces of atmosphere, land surface, ocean, lakes, and sea-ice. All this is necessary to increase the fidelity of daily forecasts and of European Copernicus Services, e.g. through the provision of state-of-the-art atmospheric monitoring services, warning systems for flood and fires, and providing reanalyses. A particular challenge arises from ensuring energy efficiency for these extreme-scale applications. This talk will comprehensively describe the steps taken towards preparing complex numerical weather predictions systems for potentially disruptive technology changes. This includes adaptation to heterogeneous architectures, accelerators and special compute units, adaptation to hierarchical memory layouts, increasing flexibility to use different numerical techniques with fundamentally different communication and computational patterns, frontier research on algorithm development for extreme-scale parallelism in time and in space, and minimising both time- and energy-to-solution. For example, a significant step towards further savings both in terms of throughput and speed-up is provided by the impact on simulations if numerical precision is selectively reduced in high resolution simulations.
Coupled problems in porous media with a focus on Biot
The key challenge in the successful utilisation of subsurface resources is the coupling of different physical processes involved. The model equations for the coupled behaviour of thermal, hydro, mechanical and chemical effects lead to a system of nonlinear, coupled, possibly degenerate PDEs.
We consider a system of coupled PDEs that incorporates the evolution in the pore-scale geometry. Starting from a reactive flow transport model describing the precipitation-dissolution processes, we visit the recent works that take into account the variation in the pore-scale geometry. Next, we consider the coupled flow and geomechanics model (Biot model) that takes into account the deformations due to the mechanical effects. Specifically, we show the iterative schemes for solving the Biot equation and the different extensions including non-linearities and further physics.
"How Warm is it Getting?" and Other Tales in Uncertainty Quantification
In the statistics community “Big Data” science is meant to suggest the combining of inferential and computational thinking.
We also speak of big data in the geosciences. However, the problems we pursue are often extreme in the number of degrees of freedom, and in many instances, non-stationary in its statistics. This usually means that we are working with sparse observational data sets, even if the number of observations is large. The Bayesian framework is a natural inferential data assimilation strategy in geosciences, to some extent because the degrees of freedom in the problem vastly outnumber observations but more critically, because the models we use to represent nature have considerable predictive power.
Looking toward the future, we expect improvements in computational efficiency and finer resolutions in models, as well as improved field measurements. This will force us to contend with physics and statistics across scales and thus to think of ways to couple multiphysics and computational resolution, as well as to develop efficient methods for adaptive statistics and statistical marginalization.
How this coupling is exploited to improve estimates that combine model outcomes and data will be described in tracking hurricanes and improving the prediction of the time and place of coastal flooding due to ocean swells. Estimating the trend of Earth’s temperature from sparse multi-scale data will be used as an example of adaptivity in time series analysis.
Other open challenges in non-stationary big data problems will be described, where progress could result from “Big Data Geoscience,” the tighter integration of geoscience, computation, and inference.