ICMS

Galerkin methods with applications in weather and climate forecasting

Mar 23, 2015 - Mar 27, 2015

ICMS, 15 South College Street, Edinburgh

Organisers

Name	Institution
Cotter, Colin	Imperial College London
Dedner, Andreas	University of Warwick
Giraldo, Frank	Naval Postgraduate School
Taylor, Mark	Sandia National Laboratories

Researchers developing, analyzing, and applying Galerkin methods with applications in weather forecasting and climate modelling  will be brought together in this workshop. The aim of the workshop is to involve mathematicians, computer scientists, meteorologists and other specialists interested in this class of methods. The focus will be on addressing the computational and analytical challenges faced by the next generation of climate and numerical weather prediction (NWP) models. This topic is currently highly relevant. Most modelling centres are in the process of making fundamental changes to their core discritizaton approaches to further improve forecast skill and provide more reliable information on climate change. New approaches are needed to make effective use of modern high-performance computers and allow for new capabilities such as unstructured grids and more realistic treatment of the Earth’s topography.

Arrangements

It is anticipated that the workshop will commence with Registration on Monday morning 23 March - and close on Friday afternoon 27 March 2015.   Please take these timings into account when organising your travel to the workshop.

Participation

• Invited participants - Invitations were emailed in November 2014.

• Other participants - A limited number of places are available at this workshop.  It is recommended that you apply as soon as possible and certainly by the closing date of 18 January 2015. The organizers will make a decision on the applications by 26 January 2015 and you will be informed by email after that date. We are able to provide accommodation in Edinburgh (for up to 5 nights) but you must pay all of your own travel costs to and from Edinburgh. A registration fee of 70.00 GBP is payable by all participants. If you would like to be considered for one of these places please complete the application form. 

Registration Fee
A 70.00 GBP registration fee is payable by all. 

• Payment can be made on arrival at ICMS - at Registration we will accept cash, credit/debit card payments and sterling cheques (payable to “Heriot-Watt University”).  Unfortunately, we do not yet have an on-line payment system at ICMS.   If you wish to pay by credit/debit card, please complete the form (available in the link below) and bring the completed form to Registration.  

Venue and Talks
The workshop will be held at 15 South College Street, Edinburgh.  You may also find this map useful for the workshop.  All lectures will be held in the Newhaven Lecture Theatre.  The Lecture Theatre is equipped with a data projector, computer, visualiser (the new generation of overhead projectors) and two blackboards.  The projector and one board may be used simultaneously.  It is best to bring your presentation on a memory stick to use in our ICMS computer.  Alternatively, it is possible for you to use your own laptop with our dataprojector, but please be aware that you may have to alter your laptop resolutions/settings.   

Travel
Information about travel to the UK and Edinburgh is available here.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies (volcanic ash disruptions) that may occur on your trip.

A taxi directly from the airport will cost approximately 16.00 to 20.00 GBP to the city centre for a one-way journey.  There is also a bus service direct from the airport to the city centre which will cost 4.00 single or 7.00 GBP return - the Airlink 100.  This is a frequent service (every 10 minutes during peak times) and will bring you close to Waverley Railway Station, only a short walk to the accommodation and the workshop venue (see map in 'Venue' section above). 

Lothian buses charge £1.50 for a single, £3.50 for a day ticket.  Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has several railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge.  The other railway stations are Haymarket and Edinburgh Park at the West End of the city centre.

UK Visas
If you are travelling from overseas you may require an entry visa.  A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one.

Accommodation
If requested, accommodation will be arranged by ICMS and details of your accommodation emailed to you nearer the time of the workshop.  

Public Lecture
On Tuesday 25 March, David Stainforth will give a Public Lecture (title and abstract below).  Doors will open to the Public at 17.30, with the Lecture beginning at 18.00.  The Public Lecture will be followed by an informal wine reception in the Chapterhouse at ICMS from 19.00-20.00.    Seats have been reserved in the Lecture Theatre for all the workshop participants so there is no need for you to book a seat.

Confidence from Uncertainty: Separating what we know from what we don’t know about climate change
The basis for expecting mankind’s emissions of greenhouse gases to change climate is founded in well understood physical principles. This basis enables us to make some statements about the risks and threats posed by climate change with very high confidence. Yet climate prediction also requires us to acknowledge uncertainties. Uncertainties in how the climate system works, uncertainties in how human society uses the scientific knowledge we have, and uncertainties resulting from intrinsic limits to knowledge. This talk will illustrate how high confidence in some aspects of climate change goes hand in hand with substantial uncertainties in others; particularly our ability to provide details of what future climate will look like at any specific location. The results of simulations with large complicated computer models of the climate system will be presented alongside illustrations of simple mathematical concepts - the butterfly effect and its lesser known cousin, the hawkmoth effect. Such results illustrate the scale of the challenge we face when trying to predict the detailed behaviour of something as complex as the earth’s climate. They also demonstrate how increasing the exploration of uncertainty can generate increased confidence in certain aspects of future climate change. How to quantify uncertainty in climate predictions is a grand challenge of science today and this is entirely consistent with overwhelming confidence in the threat to our societies posed by climate change. Responding to such knowledge is a grand challenge for our political systems.

Catering
The workshop grant will cover refreshments throughout the event, lunch on three days, an informal wine reception and a workshop dinner.

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout.  On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there will also be a couple of computers available for you to check your emails. 

Who to contact?
If you have any enquiries about the arrangements for this workshop, please email:    moira.spencer(at)icms.org.uk .

 

Programme

Monday 23 March

09:00 - 09:40	Registration & coffee in the Chapterhouse, Level1.
09:40 - 09:45	Opening Remarks.
09:45 - 10:30	Frank Giraldo (Naval Postgraduate School) NUMA: A unified continuous/discontinuous Galerkin regional/global nonhydrostatic atmospheric model. pdf. of presentation
10:30 - 11:15	Coffee/Tea in the Chapterhouse.
11:15 - 12:00	Ram Nair (National Center for Atmospheric Research) A time-split discontinuous Galerkin global nonhydrostatic dynamical core.
12:00 - 13:45	Lunch provided in the Chapterhouse.
13:45 - 14:30	Dale Durran (University of Washington) A high-order positive-definite discontinuous Galerkin method for multi-dimensional scalar transport.
14:30 - 15:15	Alexandre Ern (Université Paris-Est) Compatible discrete operator schemes. pdf of presentation
15:15 - 16:00	Coffee/Tea in the Chapterhouse.
16:00 - 16:45	Daniel LeRoux (Université Claude Bernard Lyon 1) DG discrete analyses and numerical simulations of typical waves for 2-D environmental problems.
16:45 - 17:30	Robert Kirby (Baylor University) Fast simplicial finite element algorithms via Bernstein polynomials
17:45 - 19:00	Wine reception and poster session.

 

Tuesday 24 March

09:00 - 09:45	Christiane Jablonowski (University of Michigan) High-order adaptive mesh refinement (AMR) and variable-resolution techniques for atmospheric general circulation models
09:45 - 10:30	Michal Kopera (Naval Postgraduate School) Towards large-scale adaptive simulations in NUMA
10:30 - 11:15	Coffee/Tea in the Chapterhouse.
11:15 - 12:00	Martin Schreiber (University of Exeter) Towards efficient high-dimensional simulations on dynamically adaptive Cartesian grids. pdf. of presentation   pdf. of Poster
12:00 - 13:45	Lunch provided in the Chapterhouse.
13:45 - 14:30	Hilary Weller (University of Reading) Numerical solution of the Monge-Ampere equation for mesh adaptivity on the sphere. pdf. of presentation
14:30 - 15:15	Colin Cotter (Imperial college London) Compatible finite elements for numerical weather prediction. pdf. of presentation
15:15 - 16:00	Coffee/Tea in the Chapterhouse.
16:00 - 16:45	Alexander Linke (Weierstrass Institute)  Robust discretization of the Coriolis force in the Navier-Stokes flows on  unstructured grids pdf. of presentation
18:00 - 19:45	Public lecture & informal wine reception.

 

Wednesday 25 March

09:00 - 09:45	Fred Homewood (Cray (UK) Ltd) Cray: delivering performance for compute, storage and analysis for earth sciences  pdf. of presentation
09:45 - 10:30	Andreas Mueller (Naval Postgraduate School) Strong scaling on more than 700,000 cores with the atmospheric model NUMA.
10:30 - 11:15	Coffee/Tea in the Chapterhouse.
11:15 - 12:00	Philipp Birken (Lund University) Towards fast solvers for DG methods for compressible flows. pdf. of presentation
12:00 - 12:45	Panel discussion.
12:45onwards	Free afternoon.

 

Thursday 26 March

09:00 - 09:45	James Maddison (University of Edinburgh) Mesh-adaptive quasi-geostrophic modelling.
09:45 - 10:30	Lucas Wilcox (Naval Postgraduate School) Accelerating NUMA in a performance portable way. pdf. of presentation
10:30 - 11:15	Coffee/Tea in the Chapterhouse.
11:15 - 12:00	Robert Kloefkorn (International Research Institute of Stavanger) Fully implicit discontinuous Galerkin methods for atmospheric flow.
12:00 - 13:45	Lunch provided in the Chapterhouse.
13:45 - 14:30	Tan Bui-Thanh (The University of Texas at Austin) A hybridized discontinuous Galerkin method for earth system models' dynamical cores.
14:30 - 15:15	Matthew Norman (Oak Ridge National Laboratory) A cheap, large time step, multi-moment alternative to Galerkin methods. pdf. of presentation
15:15 - 16:00	Coffee/Tea in the Chapterhouse.
16:00 - 16:45	David Ham (Imperial College London) Higher level interfaces and code generation for atmospheric simulation.
16:45 - 17:30	Jemma Shipton (Imperial College London) Compatible finite element methods for numerical weather prediction.
19:00	Workshop dinner at Blonde Restaurant,71-75 St. Leonard St, Edinburgh.

 

Friday 27 March

09:00 - 09:45	Eike Mueller (University of Bath) Multigrid preconditioners for mixed finite element discretisations of atmospheric equations of motion on extruded grids pdf. of presentation
09:45 - 10:30	Carsten Burstedde (Universität Bonn) Parallel adaptive finite element methods for flow simulation.
10:30 - 11:15	Coffee/Tea in the Chapterhouse.
11:15 - 12:00	James Shaw (University of Reading) Numerical representation of mountains in atmospheric models. pdf. of presentation
12:00 - 12:45	Mark Taylor (Sandia National Laboratories) Consistent coupling of finite element advection and finite volume transport. pdf. of presentation
12:45	Close of workshop.

 

 

Presentations:

Presentation Details
Birken, Philipp
Towards fast solvers for DG methods for compressible flows
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We consider stiff problems arising from a high order discretization of compressible flows, as modelled by the Euler or Navier-Stokes equations. The stiffness makes a time adaptive implicit treatment necessary and thus methods for solving nonlinear or linear equation systems. Hereby, an attractive option is given by so called ESDIRK methods, which are implicit Runge-Kutta methods that can be implemented as a sequence of implicit Euler steps. As for the equation systems, it turns out that compared to finite volume methods, the arising systems are much harder to solve, essentially due to the huge size of blocks in the Jacobian, compared to the finite volume case. We shortly review the alternatives of preconditioned Jacobian-free Newton-Krylov methods and nonlinear multigrid schemes with appropriate smoothers. Then we discuss possible solutions regarding preconditioners and smoothers, for example the new preconditioner ROBO-SGS.
Bui-Thanh, Tan
A Hybridized discontinuous Galerkin method for earth system models' dynamical cores.
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We present an efficient high order hybridized Discontinuous Galerkin method (HDG) that is promising for the dynamical cores at extreme-scales. In effect, the HDG framework reduces the spatial dimension by one, and hence it has much less, in fact minimal, number of coupled degree of freedoms, namely the unknown traces, to solve for as compared to standard DG methods. Once the traces are computed, the field unknowns can be recovered in parallel elementby- element independent of each other. As such, HDG method is intrinsically scalable and parallelizable, and hence well suited for current and future exascale supercomputer technologies with co-processors and accelerators. We present numerical results on various PDEs including advection equation, compressible Euler and Navier-Stokes equations, and shallow water equations.
Burstedde, Carsten
Parallel adaptive finite element methods for flow simulation
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Modern parallel algorithms for adaptive mesh refinement offer increased flexibility in tailoring numerical simulations, namely by making the number of processors freely choosable and by allowing to adapt the local resolution while the simulation is running. To be competetive, an adaptive mesh code used on a uniform mesh should have negligible overhead compared to a hardcoded uniform mesh. In practice, several challenges arise from this approach. One is to enable scalability of the distributed mesh partition and modification algorithms for dimensions 2, 2.5, and 3. Another is to develop an adaptive finite element discretization that ensures conservation and incompressibility if so desired. Finally, the CFL condition is a strong motivation to use local time stepping or implicit solves when the local resolution becomes exceedingly fine. We will give an overview on these issues and present selected working examples.
Cotter, Colin
Compatible finite elements for numerical weather prediction
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The compatible finite element framework has been shown to have many of the same desirable linear properties as the C-grid staggered finite difference methods, but allows for the use of non-orthogonal grids, flexibility to balance velocity and pressure degrees of freedom, and flexibility to increase the consistency order. I will review some of our recent results when applying compatible finite element methods to the rotating shallow water equations. I will then explain how to extend these finite element spaces to three dimensions, including: finite element version of the Charney-Phillips grid, construction of spaces on terrain following meshes, and the shallow atmosphere approximation. I will also discuss our approach to discretising the nonlinear advection and pressure gradient terms in vertical slice models and three dimensional models on the spherical annulus.
Durran, Dale
A High-Order Positive-Definite Discontinuous Galerkin Method for Multi-Dimensional Scalar Transport
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A positive definite modal discontinuous Galerkin (DG) method is developed for multi-dimensional scalar transport. The method uses a flux-corrected-transport style limiter to maintain non-negative cell-averaged masses, which is particularly well-adapted to the modal representation. Non-negativity is preserved within each cell by a multiplicative adjustment of the mass with equally spaced subcell-volumes. The values of the solution on these equally spaced subcell volumes are available for the computation of other terms in the governing equations, including those arising from physical parameterizations that are often most easily formulated for equally spaced meshes. The method is extended to higher dimensions using Strang splitting. Numerical tests show the method to be highly accurate, both preserving the spectral convergence of the underlying DG method and giving good results as features become very coarsely resolved in a sheared swirling flow. Despite the general tendency of unlimited modal DG methods to be less efficient than unlimited nodal DG methods, the new method exhibits greater computational efficiency that some recently proposed positive-definite nodal DG methods.
Giraldo, Frank
NUMA: A Unified Continuous/Discontinuous Galerkin Regional/Global Nonhydrostatic Atmospheric Model
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In this talk, we shall describe the NUMA atmospheric model, focusing in particular on its unified continuous/discontinuous (CG and DG) Galerkin numerical methods that are used to represent the spatial derivatives. We shall describe how these two methods are formulated in a unified approach and the advantages that this brings. Examples of such advantages include: 1) that we can compare both methods systematically and fairly, 2) that we can use the strength of one method to improve the other (e.g., limiters and time-integration), and 3) that we can use them to give us a sense of how useful some other algorithms could be for different numerical methods (e.g., preconditioners and adaptive mesh refinement). We will also describe the advantage that Galerkin methods can bring to the general area of atmospheric modeling, in particular, numerical weather prediction especially because Galerkin methods are perfectly suited for handling both regional and global modeling. In fact, NUMA has been designed to perform as a regional as well as a global nonhydrostatic atmospheric model. Frank Giraldo Department of Applied Mathematics Naval Postgraduate School Monterey, CA 93943 USA
Ham, David
Higher level interfaces and code generation for atmospheric simulation
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How will we implement ever-more complex numerical schemes and parametrisations for ever-more complex and fine-grained parallel hardware? How will we analyse the petabytes of data which result? In this talk I will discuss the prospects of dramatically simplifying the generation of complex high performance simulations from a mathematical specification of the discrete problem formulation. I will discuss the particular challenges of representing and expoiting the columnar structure of atmospheric models, and of representing operations - such as parametrisations - which do not directly fit the pure PDE abstraction. Time permitting, I will further discuss the beginnings of work to automate the analysis of atmospheric simulation data. The principle example I will consider is the Firedrake automated finite element system, however I will also illustrate lower-level compile time systems such as that being developed for the Met Office Gung Ho project.
Homewood, Fred
Cray: Delivering performance for Compute, Storage and Analysis for Earth sciences
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From it's beginning with the Cray-1 in 1976, supercomputer performance has increased by 3 x 10^9. These early computers were used by meteos the world over. Cray continues to deliver performance to Earth sciences today, this talk looks at some of the technology, components, speeds and feeds are able to achieve that speed up for real applications
Jablonowski, Christiane
High-Order Adaptive Mesh Refinement (AMR) and Variable-Resolution Techniques for Atmospheric General Circulation Models
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The paper reviews two approaches to high-order variable-resolution modeling that have recently been designed for atmospheric General Circulation Models. The first approach is based on the AMR library Chombo that supports fourth-order finite volume methods for block-structured adaptive meshes on cubed-sphere grids. The Chombo-AMR model has been jointly developed by the Lawrence Berkeley National Laboratory and the University of Michigan. The second variable-resolution approach is based on the Spectral Element (SE) method that has been implemented on a cubed-sphere grid in the Community Atmosphere Model (CAM). The latter has been jointly developed by NCAR and various Department of Energy laboratories. The paper discusses the characteristics of both variable-resolution methods using a hierarchy of test cases and flow scenarios. In particular, the AMR-Chombo model is evaluated in the 2D shallow-water framework, and various refinement criteria are compared. The CAM-SE model is assessed in a dry 3D dynamical core mode as well as in aqua-planet and realistic configurations. Special attention is paid to the flow conditions in the grid transition regions that have the potential to exhibit grid imprinting. It is shown that the high-order numerical methods successfully suppress spurious noise without the need for special diffusive mechanisms in the transition zones.
Kirby, Robert
Fast simplicial finite element algorithms via Bernstein polynomials
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For many years, sum-factored algorithms for finite elements in rectangular reference geometry have combined low complexity with the mathematica power of high-order approximation. However, such algorithms rely heavily on the tensor-product structure inherent in the geometry and basis functions, and achieving similar efficiency on triangular or tetrahedral meshes has proven elusive. Over the last several years, however, we have learned many remarkable properties of the Bernstein polynomials that, when used as finite element shape functions, admit optimal-complexity algorithms for the assembly and application of finite element operators. They also form natural building blocks in the finite element exterior calculus for bases for the de Rham complex so that H(div) and H(curl) have efficient bases as well. Very preliminary work also suggests that they may be equally efficient for discontinuous Galerkin discretization of hyperbolic problems.
Kloefkorn, Robert
Fully Implicit Discontinuous Galerkin Methods for Atmospheric Flow.
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In this talk we will discuss fully implicit time discretizations for Discontinuous Galerkin methods applied to atmospheric flow. In particular, we consider Jacobian-free Newton-Krylov solvers based on a matrix-free implementation strategy. We will present some ideas on how to incorporate proper preconditioning for the Krylov solver into this framework.
Kopera, Michal
Towards large-scale adaptive simulations in NUMA
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In this talk we will present the implementation of the adaptive mesh refinement to the unified continuous/discontinuous Galerkin (CG/DG) methods in Non-hydrostatic Unified Model of the Atmosphere (NUMA). As a measure for comparing the non-conforming DG and CG methods we chose the mass conservation error. We will show that those two method conserve to machine precision and outline the issues that arose during the study, namely an increased stabilization requirements for the CG method while extending the simulation time beyond typical test case values. We will also discuss parallel grid creation in NUMA using the p4est library and progress towards a column-based and full 3D dynamic adaptive mesh refinement.
LeRoux, Daniel
DG discrete analyses and numerical simulations of typical waves for 2-D environmental problems
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The shallow-water system, is extensively used in environmental studies (ocean, atmosphere, rivers) to simulate the propagation of typical waves (e.g. gravity, Rossby). For a number of finite-element discretization schemes, the numerical approximation of shallow-water models is a delicate problem leading to the appearance of non physical elevation and inertial solutions and dispersion/dissipation problems in the representation of the waves. In order to understand these difficulties and to select appropriate spatial discretization schemes, Fourier analyses have proven beneficial. The aim of this talk is to present such results in the context of discontinuous Galerkin methods, and to propose a class of possible discretization schemes that is not affected by the spurious modes. Both discrete analyses and numerical simulations, involving wave propagation for environmental problems, will be presented for the 1-D and 2-D cases.
Linke, Alexander
Robust discretization of the Coriolis force in Navier-Stokes flows on unstructured grids
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About fifty years ago, numerical mathematicians realized that a certain surjectivity property of the discrete divergence is needed for asymptotically optimal convergent discretizations of the Navier-Stokes equations. However, numerical mathematicians did not realize that the discrete divergence constraint not only induces a discrete divergence (via the trial functions), but also a discrete rotation operator (via the test functions). As a consequence, they constructed mixed flow discretizations that fulfil the vector calculus property 'gradient fields are irrotational' only approximately. Such flow discretizations necessarily fail for flow problems with large irrotational forces in the momentum balance, as it appears for many flows in meteorology, where the Coriolis force is comparably large. Therefore, we propose novel mixed discretization schemes for the Navier-Stokes equations in 2d and 3d that are able to treat the Coriolis force in a robust way, since they fulfil exactly 'gradient fields are irrotational'. Numerical examples demonstrate the theoretical findings.
Mueller, Andreas
Strong scaling on more than 700,000 cores with the atmospheric model NUMA.
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Even with today's supercomputers it is still impossible to explicitly represent all scales involved in weather prediction. A large number of different numerical methods have been developed for improving the efficiency of numerical simulations. Among those methods are the element-based Galerkin methods which allow arbitrary high order for the spatial discretization and promise excellent scalability on large supercomputers. The authors are developing the dynamical core NUMA which is used within the next-generation weather prediction model NEPTUNE of the US Navy. NUMA incorporates many recent developments like spectral element and discontinuous Galerkin methods. Recently, the authors have shown strong scaling of NUMA on more than 700,000 cores using more than 1.5 million MPI threads. The scalability study was performed on the supercomputer Mira of the Argonne National Laboratory using the spectral element method. Two test cases are considered: 1) a baroclinic instability test case by Jablonowski and Williamson (2006) on the sphere using a cubed sphere mesh and 2) a 3D rising thermal bubble test case in a cube. Details of this scalability study are presented and the sensitivity of the results with respect to the polynomial order of the spatial discretization is discussed. Authors: Andreas Mueller, Michal Kopera, Simone Marras, Francis X. Giraldo.
Mueller, Eike
Matrix-free multigrid preconditioners for mixed finite element discretisations of atmospheric equations of motion on extruded grids
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(Mimetic-) mixed finite element discretisations are popular in atmospheric modelling since they allow the conservation of physical quantities on unstructured grids. For forecast models which use implicit time stepping methods, a linear system of equations for the atmospheric variables has to be inverted in every time step. To solve the finite element discretisation of this system efficiently, a suitable preconditioner has to be employed to calculate the pressure correction in a discontinuous Galerkin space. We developed a bespoke matrix-free preconditioner with a geometric multigrid solver for the pressure correction equation; vertical line relaxation is used to handle the strong vertical anisotropy in flat domains. Since the velocity mass matrix is not diagonal, the algorithm is significantly more complex than the corresponding finite volume version and requires careful implementation, testing and tuning. The solver was implemented in the performance-portable firedrake/PyOP2 framework which allows the easy expression of the algorithm at a high abstraction level. We tested the performance of the solver for a linarised dimensional gravity wave system in a thin 2+1 dimensional spherical shell, which is realised as an extruded mesh. Results are reported for the algorithmic- and parallel scalability both for lowest-order- and higher-order finite element discretisations
Nair, Ram
A Time-Split Discontinuous Galerkin Global Nonhydrostatic Dynamical Core
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A global nonhydrostatic (NH) model based on compressible Euler equations has been developed in the HOMME framework. The spatial discretization is based on a nodal discontinuous Galerkin (DG) method. The 3D NH model is designed to be suitable for operator-split approaches. The horizontal discretization of the model relies on cubed-sphere geometry, employing 2D rectangular elements, and the vertical discretization is based on the terrain-following height-based 1D elements to handle topography. The time integration uses the horizontally explicit and vertically implicit (HEVI) time-splitting scheme, which is introduced to address the stringent restriction on the explicit time step size due to a high aspect ratio between the horizontal and vertical spatial discretization. The HEVI scheme is generally based on the Strang-type operator-split approach, where the horizontally propagating waves in the Euler system are solved explicitly while the vertically propagating waves are treated implicitly; and the time-step of the model is only limited by the horizontal CFL. A subset of the DCMIP (Dynamical Core Model Inter-comparison Project) tests is used to validate the accuracy of the model. The HEVI approach used for the NH has the potential to maintain the petascale capability of the HOMME framework
Norman, Matthew
A Cheap, Large Time Step, Multi-Moment Alternative to Galerkin Methods
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Galerkin methods exhibit excellent accuracy owed to the locality of evolved DOFs and also minimal parallel communication per time step. The time step, however, reduces nearly quadratically with increasing order, handicapping high-order methods for high-throughput sciences like climate and weather. This talk introduces Boundary-Averaged Multi-moment Constrained finite-Volume (BA-MCV) schemes, which correct this. For the same order of accuracy, they are slightly less accurate than Spectral Element (SE) operators, yet they maintain a maximum stable CFL value of one regardless of order. They are well-behaved at discontinuities without explicit limiting. They are also easily limited with Weighted Essentially Non-Oscillatory (WENO) limiters or hyperviscosity. They cost significantly less that SE due to the time step and expense per time step. This is vital for increasing the threading per node for accelerated architectures. The BA-MCV method evolves the cell average and pointwise derivatives of the state at cell boundaries. The average is evolved with a straightforward FV method, giving exact mass conservation, and the pointwise derivatives are evolved in strong form via derivatives of the PDE. An Nth-order-accurate BA-MCV method only transfers (N+1)/2 values per cell boundary per time step in parallel in 1-D, providing excellent scalability.
Schreiber, Martin
Towards efficient high-dimensional simulations on dynamically adaptive Cartesian grids
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Solving PDEs on Cartesian grids is a non-trivial task once dynamic adaptivity is required. Whereas space-filling curves (SFC) already proofed to be an efficient way for a frequent repartitioning and load-balancing of the underlying grid, one of the still-existing challenges is the requirement of efficient data exchange with adjacent grid cells under the circumstances of a dynamically changing grid. In this work, our first focus is on the comparison of indexed- with stack-based connectivity methods for DG simulations of shallow-water equations on Cartesian grids. Here, the cell access order is given by the Z, Hilbert & Peano SFCs. We show implications for indexed schemes regarding hanging nodes and, based on previous work for two-dimensional simulations, present an elegant solution based on the stack-communication approach. Then, we present certain challenges for the stack-based communication arising for simulations on higher-dimensional dynamically adaptive grids and their solution with the Z and Peano SFC. This is followed by first results for this connectivity scheme applied to three and higher dimensions. Our work can used with lower-dimensional grids, but we see the main potential in application scenarios demanding for higher-dimensional grids: E.g. 4 dimensions for a parallelization in time with a dynamical-adaptive time discretization, 6 dimensions for the Boltzmann equations and arbitrary dimensions for big data scenarios. Our previous work on stack-based communication schemes shows the possible extension to an efficient cluster-based parallelization strategy, resulting in several beneficial advantages such as the support of shared- and distributed-memory parallelization models as well as cluster-based optimizations.
Shaw, James
Numerical Representation of Mountains in Atmospheric Models
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The most common representation of terrain in atmospheric models is that of terrain following layers. An alternative is the cut cell method, which has been found to give more accurate results in a limited number of test cases from the literature. This project assesses the accuracy of terrain following and cut cell style grids, using the same model to enable like-for-like comparison between grids. We find that terrain following grids give better accuracy when simulating orographically induced gravity waves, with the Lorenz computational mode giving rise to spurious thermal anomalies on the cut cell style grid. This motivates the formulation of a Charney–Phillips staggering for unstructured grids in which potential temperature values are stored at cell faces instead of cell centres.
Shipton, Jemma
Compatible finite element methods for numerical weather prediction.
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We present benchmarking results from using mixed finite element methods that satisfy the conditions of finite element exterior calculus to solve the nonlinear shallow water equations on a rotating sphere. These methods share the desirable mimetic properties exhibited by C-grid finite difference schemes, namely energy conservation, mass conservation, no spurious pressure modes and steady geostrophic modes, but without orthogonality constraints on the grid. These methods are currently being developed as part of the “Gung Ho” UK dynamical core project. We will present details of the timestepping method, including stable, consistent advection schemes for both the layer depth (discontinuous) and potential vorticity (continuous) Galerkin fields. These schemes lead to a numerically conserved diagnostic potential vorticity, essential for accurate representation of predominately balanced flows. We demonstrate the accuracy and order of convergence of these schemes using standard test cases. Authors: J. Shipton. and C. J. Cotter, Department of Mathematics, Imperial College, London
Taylor, Mark
Consistent coupling of finite element advection and finite volume transport
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I will describe our work integrating the CSLAM flux form characteristic based (semi-Lagrange) transport algorithm into the HOMME spectral finite element atmospheric dynamical core. A key issue is in obtaining mass and tracer mass consistency with respect to the CSLAM control volumes, which are subcells of each spectral finite element. For the flux form CSLAM, this requires advecting the tracers with a mean mass flux from the dynamics. For finite element methods which obtain local conservation through mimetic properties of divergence and gradient discretizations, fluxes are never computed. We show that for mimetic finite element methods one can derive an implied edge flux for the CSLAM control volumes. That is, if one computes the change in mass within the CSLAM control volumes from the dynamics' finite element representation of densiy, this change in mass is given by the sum of the implied edge fluxes. The edge fluxes are equal and opposite on shared edges, ensuring local conservation. These edge fluxes can be computed locally, with no additional communication. When advecting tracers with these edge fluxes, tracer mass will remain consistent with mass computed by the dynamics.
Weller, Hilary
Numerical Solution of the Monge-Ampere equation for mesh adaptivity on the sphere
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Mesh redistribution, or r-adaptivity, is an attractive form of adaptivity since mesh connectivity does not need to change, mapping between different meshes is not needed and mesh adaptivity does not lead to parallel load imbalance, as occurs for h-adaptivity during insertion and deletion of points. Optimal transport, solving a Monge-Ampere equation, is an excellent technique to redistribute a mesh since it avoids mesh tangling. A numerical technique for solving the Monge-Ampere equation on a sphere for global atmospheric r-adaptivity will be described and results on r-adaptive meshes presented.
Wilcox, Lucas
Accelerating NUMA in a Performance Portable Way
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The ever increasing pace of improvement in the state-of-the-art high performance computing technology promises enhanced capabilities for the next-generation atmospheric models. The challenge it brings is the need for optimizing and tuning of code for optimal performance on current and future architectures. In this talk we discuss the challenges and success we have had while implementing various numerical algorithms from the Nonhydrostatic Unified Model of the Atmosphere <http://faculty.nps.edu/fxgirald/projects/NUMA>, also known as NUMA, on many-core machines. Our implementation is designed to fully exploit the current and future generations of parallel many-core computers in a performance portable way. We will discuss how we achieved threading platform portability through OCCA2 <https://github.com/tcew/OCCA2> and our plans and experience for performance portability using Loopy <http://mathema.tician.de/software/loopy>.

Participants

Name	Institution
Benacchio, Tommaso	Met Office
Birken, Philipp	Lund University
Bui-Thanh, Tan	The University of Texas at Austin
Burstedde, Carsten	Universität Bonn
Cotter, Colin	Imperial College London
Dedner, Andreas	University of Warwick
Durran, Dale	University of Washington
Erath, Christoph	TU Darmstadt
Ern, Alexandre	Université Paris-Est
Giraldo, Frank	Naval Postgraduate School
Ham, David	Imperial College London
Homewood, Fred	Cray (UK) Ltd
Jablonowski, Christiane	University of Michigan
Kim, Mi-Young	Inha University
Kirby, Robert	Baylor University
Kloefkorn, Robert	International Research Institute of Stavanger
Kopera, Michal	Naval Postgraduate School
Kritsikis, Evaggelos	Universite Paris 13
LeRoux, Daniel	Université Claude Bernard Lyon 1
Linke, Alexander	Weierstrass Institute
Maddison, James	University of Edinburgh
Mitchell, Lawrence	Imperial College London
Mueller, Andreas	Naval Postgraduate School
Mueller, Eike	University of Bath
Nair, Ram	National Center for Atmospheric Research
Norman, Matthew	Oak Ridge National Laboratory
Peddle, Adam	University of Exeter
Peixoto, Pedro	University of Exeter / University of Sao Paulo
Pring, Stephen	Met Office
Schreiber, Martin	University of Exeter
Shaw, James	University of Reading
Shipton, Jemma	Imperial College London
Stainforth, David	London School of Economics and Political Science
Taylor, Mark	Sandia National Laboratories
Weller, Hilary	University of Reading