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Transcript of M. Baldauf (DWD)1 29 th WGNE meeting 10-14 March 2014, Melbourne M. Baldauf (DWD) Recent...
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M. Baldauf (DWD) 1
29th WGNE meeting 10-14 March 2014, Melbourne
M. Baldauf (DWD)
Recent developments in Numerical Methods -with a report from the ECMWF annual seminar on ‚Recent developments in numerical methods for atmosphere and ocean modelling 2013‘
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M. Baldauf (DWD) 2
ECMWF annual seminar 2013 on ‚Recent developments in numerical methods for atmosphere and ocean modelling‘
2-5 Sept. 2013, ECMWF, Reading, UK
organizer: Nils Wedi (ECMWF)
invited talks also from many globally modelling centershttp://www.ecmwf.int/newsevents/meetings/annual_seminar/2013/index.html(last annual seminars on numerical methods: 2004, 1998, 1991, …)
horizontal grid: N. Wood, G. Zängl, B. Skamarock, H. Tomita, J. Thuburnvertical discret.: G. Zängl, M. Hortal, M. BaldaufFinite element based: C. Cotter, F. Giraldo, M. Baldauftime integration: P. Benard, R. Klein, S.-J. Lockunstructured meshes: J. Szmelterreduced equation systems: P. Smolarkiewicz, R. KleinPhysics-dynamics-coupling: S. Malardel
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M. Baldauf (DWD) 3
Classical requirements for a dynamical core:
•Stability
•Accuracy (up to a certain order; at least 2)
•Efficiency
Today, additional requirements:
•conservation of certain variables
•„mimetic“ properties (discretization reproduces some exact analytical properties)
•well-balancing
•scalability
•efficient on (quite) different computer architectures (CPU, GPU,…)
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M. Baldauf (DWD) 4
Development and projection of HPC over the years (source: www.top500.org)
factor 1.8 per year= factor 2 per 14 months
estimated~108 cores
code scalability is crucial
#1
#500
sum over all 500
199
3
200
3
201
3
1 GFlops
1 TFlops
1 PFlops
1 EFlops
3*106 cores
~104 cores
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M. Baldauf (DWD) 5
Since the clock rate of supercomputers does not increase
increase of computer power only by parallelisation
scalability of the code is very important
Many current global models (still) use a lat-lon grid
clustering of grid points around the pole
get rid of the strong time step restriction by combination of Semi-Lagrangian-advection and semi-implicit time integration
SL for high Courant numbers requires intensive communication
detrimental for scalability
look for horizontally more uniform grids(lat-lon is not a problem for limited area models)
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M. Baldauf (DWD) 6
Horizontal grids
review of horizontal grids on the sphere: Staniforth, Thuburn (2012) QJRMS
Why bothering with all this horizontal grid stuff?
•more uniform grids (see above)•grid staggering• degrees of freedom (DoFs) per cell and/or variable
see also a general overview about ‚computational modes‘ by J. Thuburn
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Frequency Phase-, group-velocity
unstaggered
staggered
grid
Dispersion relation of the linear 1D wave equation
vph
vgr
vph
vgr
k x
k x
k x
k x
x/c
x/c
=0 for 2x waves
tstagg = ½ tunstagg
negative groupvelocity
M. Baldauf (DWD) 8
Why grid staggering?
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M. Baldauf (DWD) 9N. Wood (UK MetOffice)
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M. Baldauf (DWD) 10N. Wood (UK MetOffice)
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M. Baldauf (DWD) 11N. Wood (UK MetOffice)
unfortunately low accuracy of discretizations
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M. Baldauf (DWD) 12
B. Skamarock(NCAR)
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M. Baldauf (DWD) 13
Currently used grids for operational models
•UK MetOffice: currently lat-lon grid (‚New Dynamics‘, ‚ENDGame‘)GungHo: more uniform grids (example see above)
•NCAR: MPAS uses icosahedron dual grid hexagonal C-grid
•DWD: currently (GME): icosahedron (however, regular grid on 10 diamonds)new: ICON uses icosahedron with triangle refinement, triangular C-grid
•Canada: currently lat-lonplans: Yin-Yang grid in 2015
•China (CMA): FD on regular lat-lon grid SISL schemenew: (multi-moment) FD; Yin-Yang grid or cubed sphere.
•…
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M. Baldauf (DWD) 14
B. Skamarock(NCAR)
this is still a conformal grid(i.e. no hanging nodes)
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M. Baldauf (DWD) 15
• ECMWF: horizontally spectral (vertically FE)
• MeteoFrance: currently spectral models (share dyn. core with ECMWF)however, vision for FE discretization horizontally
• JMA: GSM uses spectral approach
• Brazil: spectral / lat-lon SI integration
• …
alternatively: hide most of the problems with the horizontal grid behind a spectral representation
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16
The Integrated Forecasting System (IFS)
technology applied at ECMWF for the last 30 years …
A spectral transform, semi-Lagrangian, semi-implicit (compressible) (non-)hydrostatic model
Big obstacle could be removed:Legendre transform is O(N^3), fast Legendre Transform O(N^2 log N)(Wedi et al. (2013), Tygert (2008, 2010) )
Similar statements hold for Aladin (biperiodic Fourier series)
N. Wedi (ECMWF)
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M. Baldauf (DWD) 17
N. Wedi (ECMWF)
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M. Baldauf (DWD) 18
by G. Mozdzynski
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M. Baldauf (DWD) 19
Vertical grids
issues:
•vertical finite element representation
•proper vertical averages in vertically staggered grid for FD are necessary
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Improvement of the vertical discretization
Arithmetic average from half levels to main level:
Weighted average from main levels to half level
Derivatives always by centered differences (appropriate average used before)
M. Baldauf (DWD) 20
Vertical (Lorenz)-staggering in COSMO (and a few other models):Half levels (w-positions) are defined by a stretching function zk = f(k); Main levels (p‘, T‘-pos.) lie in the middle of two half levels
The above mentioned staggering requires proper averaging; example …
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Divergence with weighted average of u (and v) to the half levels:
Divergence with only arithmetic average of u (and v) to the half levels:
Divergence – grid stretching variant B
not a consistent discretization if s1 !
1/s ·dz
dz
s·dz
Truncation error in stretched grids
M. Baldauf (DWD) 21
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M. Baldauf (DWD) 22
Time integration
•Meteo France (P. Benard): 3-time level Semi-implicit Leapfrog based integration+ iteration step (ICI-scheme) (on spectral model)
•DWD (ICON): 2-time level Predictor-Corrector –scheme (HEVI in each step)COSMO: 2-timelevel Runge-Kutta (split explicit)
•NCAR: 2-timelevel Runge-Kutta (split explicit) for WRF and MPAS
•UK MetOffice: currently SI-SL schemeoutlook: IMEX Runge-Kutta under consideration
•JMA: GSM uses SI-SL scheme (for the spectral model)
•…
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M. Baldauf (DWD) 23
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M. Baldauf (DWD) 24
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M. Baldauf (DWD) 25
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M. Baldauf (DWD) 27
Advection schemes
For tracer advection beyond sufficient accuracy, two ingredients are seen as important:
•conservation finite volume discretization
•consistency with continuity equation (=the advection equation for q should reproduce those of if q=1)easy, if the same advection scheme is used for as for all tracers; however, NICAM, ICON, …use different time steps for continuity equation and tracers use time averaged mass fluxes! (talks by H. Tomita, G. Zängl)
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M. Baldauf (DWD) 28
Semi-Lagrangian advection (overview given by M. Diamantakis)as used for the dynamical core in:•ARPEGE(MeteoFrance)/•IFS(ECMWF)/ALADIN•UM(UKMO)•HIRLAM•SL-AV(Russia)•GEM(Environment Canada)•GFS(NCEP)•GSM(JMA)•…
For tracer advection, local conservation properties are not easy to achieve:e.g. SLICE-3D (Zerroukat, Allen (2012) QJRMS) implemented in ENDGame version of UM (but computationally expensive)
on unstructured grids: conservative SL probably too expensive.
(conservative) SL schemes on structured grids may be beneficial for many tracers
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M. Baldauf (DWD) 29
Finite-element based methods
several Continuous Galerkin (CG) / spectral elements /Discontinuous Galerkin (DG) developments:
•NUMA (Naval postgraduate school, Monterey) F. Giraldo, …•HOMME (NCAR) R. Nair, …•CAM-SE (Sandia Nat. lab.) M. Taylor•COSMO (DWD) first steps by D. Schuster, …•…
general remarks in talk of C. Cotter
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M. Baldauf (DWD) 30
C. Cotter (Imperial Coll.)linear 1D wave equation
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M. Baldauf (DWD) 31
C. Cotter (Imperial Coll.)
Ladyzhenskaya/Babuska/Brezzi (LBB) condition guarantees compatibility
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Discontinuous Galerkin (DG) methods in a nutshell
From Nair et al. (2011) in ‚Numerical techniques …
weak formulation(increases solution space)
Finite-element ingredient
Finite-volume ingredient
ODE-system for q(k)jl
Lax-Friedrichs flux
M. Baldauf (DWD) 32
e.g.Cockburn, Shu (1989) Math. Comput.Cockburn et al. (1989) JCP
e.g. Legendre-Polynomials
talk by F. Giraldo
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1D wave expansion with a Discontinuous Galerkin (DG) discretization
Literature:
Hu, Hussaini, Rasetarinera (1999) JCP: 1D advection-, 2D wave-equation
Hu, Atkins (2002) JCP: non-uniform grids k=k()
Ainsworth (2004) JCP: multi-dim. advection equation
M. Baldauf (DWD) 33
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DG with p=0,1,2,3(=c used)
k x
k x
Re x/c
Im x/c
p max ||x/c
0 1
1 3.9
2 7.51
3 11.83
4 16.86
5 22.58
6 28.96
10 60.75
15 113.68
max || x/c 1 + 2.6 p + 0.33 p²increases slightly stronger than linear with p.Choose not too large p!
M. Baldauf (DWD) 34
Numerical dispersion relation for the 1D wave equation
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M. Baldauf (DWD) 35
F. Giraldo (NPS)
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M. Baldauf (DWD) 36
Other remarks
•„WGNE table about computer ressources 2014“up to now, I have received contributions fromChiashi (Japan), Hoon (Korea), Xueshun (China), Ayrton (Canada),Jean-Noel (ECMWF) and DWD
•shall we continue the survey about dynamical cores (by Mikhail Tolstyk) ?