Dynamical Core Developments at KIAPS

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1 Dynamical Core Developments at KIAPS 1/ 48 Dynamical Core Developments at KIAPS Preliminary results T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Korea Institute of Atmospheric Prediction Systems 24 September, 2012 / PDEs on the sphere T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 1/ 48

2 Dynamical Core Developments at KIAPS 2/ 48 Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 2/ 48

3 Dynamical Core Developments at KIAPS 3/ 48 Acknowledgments Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 3/ 48

Dynamical Core Developments at KIAPS 4/ 48 Acknowledgments Special Thanks To Dept. of Applied Mathematics, Naval Postgraduate School Prof. Francis X. Giraldo Dr.

4 Dynamical Core Developments at KIAPS 4/ 48 Acknowledgments Special Thanks To Dept. of Applied Mathematics, Naval Postgraduate School Prof. Francis X. Giraldo Dr. Michal Kopera Dr. Shiva Gopalakrishnan Dr. Lester Carr III T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 4/ 48

5 Dynamical Core Developments at KIAPS 5/ 48 KIAPS Introduction Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 5/ 48

Dynamical Core Developments at KIAPS 6/ 48 KIAPS Introduction KIAPS Introduction Brief Introduction KIAPS - Korea Institute of Atmospheric Prediction Systems 9 Year

Young-Joon Kim (formerly from NRL) appointed as Director (16 March, 2012) Vision & Goals To develop a global NWP system optimized to the topographic & meteorological

6 Dynamical Core Developments at KIAPS 6/ 48 KIAPS Introduction KIAPS Introduction Brief Introduction KIAPS - Korea Institute of Atmospheric Prediction Systems 9 Year project, approx. $100M funded by South Korean Government Dr. Young-Joon Kim (formerly from NRL) appointed as Director (16 March, 2012) Vision & Goals To develop a global NWP system optimized to the topographic & meteorological features of Korean Peninsula To reduce the economic loss caused by natural disasters and enhance productivity of industrial sector To build science & technology capacity that stimulates NWP research T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 6/ 48

7 Dynamical Core Developments at KIAPS 7/ 48 KIAPS Introduction KIAPS 9 Year Roadmap T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 7/ 48

8 Dynamical Core Developments at KIAPS 8/ 48 KIAPS DyCore Design Considerations Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 8/ 48

9 Dynamical Core Developments at KIAPS 9/ 48 KIAPS DyCore Design Considerations Supercomputing Trend 1 Roughly x1000 computation power increase in 10 years Clock speed Core count (heat problem) Scalability will become one of the key issues in NWP modeling 1 June 2012 data, KMA HaeDam 55th, HaeOn 56th, KISTI Tachyon II 64th, ( T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 9/ 48

10 Dynamical Core Developments at KIAPS 10/ 48 KIAPS DyCore Design Considerations UM Scalability 2 Figure : Global UM Scalability 2 Global (N512), 2011 Brown & Wood UKMO MOSAC (ppt) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 10/ 48

11 Dynamical Core Developments at KIAPS 11/ 48 KIAPS DyCore Design Considerations DyCore Requirements Global, Nonhydrostatic Model Highly scalable without pole singularity on the grid structure Local numerical procedures (minimum communication) Mass conservation High order accuracy Computationally Efficient (i.e., that satisfies operational cutoff time) Adaptive Mesh Refinement (AMR) capability (optional) In particular, these requirements eliminate the usage of... Lat-Lon grid Global Spectral method T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 11/ 48

12 Dynamical Core Developments at KIAPS 12/ 48 KIAPS DyCore Design Considerations KIAPS DyCore Design Options 1 Option 1 2 Option 2 3 Option 3 Icosahedral /SCVT Finite Volume NCAR MPAS, NOAA FIM/NIM, DWD/MPI-M ICON, CCSR/JAMSTEC NICAM Cubed sphere Spectral Element (SE, CG for Continuous Galerkin) /Discontinuous Galerkin NCAR CAM-SE, NPS NUMA Yin-Yang grid Finite Volume CMC-GEM T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 12/ 48

13 Dynamical Core Developments at KIAPS 13/ 48 KIAPS DyCore Design Considerations CAM-SE 3 Scalability Figure : CAM-SE Scalability 3 Dennis et al. (2012) CAM-SE, CESM-atm component with full physics T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 13/ 48

14 Dynamical Core Developments at KIAPS 14/ 48 KIAPS DyCore Design Considerations NUMA 4 Scalability Figure : NUMA Scalability 4 Kelly and Giraldo (2012) NUMA, Rising thermal bubble problem, 32 3 elements and N=8 T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 14/ 48

15 Dynamical Core Developments at KIAPS 15/ 48 KIAPS DyCore Design Considerations CG/DG 5 and FV 6 Communication Comparison Figure : CG vs DG Communication Requirements Figure : FV/FD vs CG/DG Computational Stencil 5 Giraldo (2012) Cubed-Sphere and Galerkin Approaches Slide by Ram Nair T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 15/ 48

Dynamical Core Developments at KIAPS 16/ 48 KIAPS DyCore Design Considerations CG/DG Properties Shares many common features Highly accurate Interpolation, Derivative, Quadrature calculation possible

16 Dynamical Core Developments at KIAPS 16/ 48 KIAPS DyCore Design Considerations CG/DG Properties Shares many common features Highly accurate Interpolation, Derivative, Quadrature calculation possible Quadrature point clustering near element boundary - no free lunch Nodal or Modal basis function Conservative Local operation High-order accurate Excellent scalability Grid Flexibility (equi-distant, orthogonality not required) Straightforward boundary condition implementation procedure T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 16/ 48

17 Dynamical Core Developments at KIAPS 17/ 48 KIAPS DyCore Design Considerations CG/DG Properties (cont.) CG DG C 0 Continuity Communication by DSS Stabilization strategy required (hyper-viscosity, modal filtering) Element-wise Discontinuity Communication by flux calculation Choice of numerical flux method is important T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 17/ 48

18 Dynamical Core Developments at KIAPS 18/ 48 1D CG/DG Advection, SWE Results Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 18/ 48

19 Dynamical Core Developments at KIAPS 19/ 48 1D CG/DG Advection, SWE Results CG/DG High Order Convergence Study T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 19/ 48

20 Dynamical Core Developments at KIAPS 20/ 48 1D CG/DG Advection, SWE Results CG/DG High Order Convergence Study (cont.) /Inexact, Np [ ], N [ ] N1 Inexact 17th order 5th order N1 Exact DG 9th order N1 Inexact N1 Exact DG N1 Exact CG N 4 Inexact N4 Exact N1 Exact CG N 1 CG Inexact N 1 DG Inexact N 1 CG Exact N 1 DG Exact N 4 CG Inexact N 4 DG Inexact N 4 CG Exact N 4 DG Exact N 8 CG Inexact N 8 DG Inexact N 8 CG Exact N 8 DG Exact N 16 CG Inexact N 16 DG Inexact N 16 CG Exact N 16 DG Exact N 16 group 6 N 8 group 7 N 4 Inexact T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 20/ 48

21 Dynamical Core Developments at KIAPS 21/ 48 1D CG/DG Advection, SWE Results High Order Convergence Requisite - Time/Space Order Balance To achieve theoretical high order convergence rates, spatial and temporal discretization orders should match Time Integrators that exceed 4th order (e.g., RK4) is rarely needed for FV methods as most schemes are less than 4th order For CG/DG, getting arbitrary spatial high order schemes is trivial Time Integration is the bottleneck. Using Gottlieb (2005), we implemented generalized m stage m 1 order Strong Stability Preserving (SSP) explicit RK time integration T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 21/ 48

22 Dynamical Core Developments at KIAPS 22/ 48 1D CG/DG Advection, SWE Results M Stage M 1 SSP-RK TI Convergence rate T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 22/ 48

23 Dynamical Core Developments at KIAPS 23/ 48 1D CG/DG Advection, SWE Results High Order Convergence Requisite - Initial Condition Cosine bell: locally definable IC C 2 discontinuous Difficult to achieve high order convergence (use Gaussian instead) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 23/ 48

24 Dynamical Core Developments at KIAPS 24/ 48 1D CG/DG Advection, SWE Results High Order Convergence Requisite - Initial Condition (cont.) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 24/ 48

25 Dynamical Core Developments at KIAPS 25/ 48 1D CG/DG Advection, SWE Results High Order Convergence Requisite - Initial Condition (cont.) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 25/ 48

26 Dynamical Core Developments at KIAPS 26/ 48 1D CG/DG Advection, SWE Results 1D ADV Convergence Rate T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 26/ 48

27 Dynamical Core Developments at KIAPS 27/ 48 1D CG/DG Advection, SWE Results 1D ADV Variable Resolution Convergence Rate T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 27/ 48

28 Dynamical Core Developments at KIAPS 28/ 48 1D CG/DG Advection, SWE Results Variable Resolution Wall Clock 7 (reg vs clustered) 7 MATLAB serial code used T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 28/ 48

29 Dynamical Core Developments at KIAPS 29/ 48 1D CG/DG Advection, SWE Results Accuracy/Wall Clock Analysis (reg) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 29/ 48

30 Dynamical Core Developments at KIAPS 30/ 48 1D CG/DG Advection, SWE Results 1D SWE Convergence Rate T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 30/ 48

31 Dynamical Core Developments at KIAPS 31/ 48 1D CG/DG Advection, SWE Results 1D SWE Convergence Rate (cont.) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 31/ 48

32 Dynamical Core Developments at KIAPS 32/ 48 2D Global SWE Results Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 32/ 48

33 Dynamical Core Developments at KIAPS 33/ 48 2D Global SWE Results Governing Equations Following Giraldo (2001) The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapping takes into account the curvature of the high-order spectral elements, thereby allowing the elements to lie entirely on the surface of the sphere µ is a Lagrange multiplier, required to constrain the fluid particles to remain on the surface of the sphere (Côté, 1988) t φ φu φv φw + x = φu φu 2 φuv φuw φ φ x + f ( z φ φ y + f ( x φ φ z + f ( y + y 0 φv φuv φv 2 φvw + z φv y φw) + µx a φw a z φu) + µy a a φu x φv) + µz a a φw φuw φvw φw 2 T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 33/ 48

34 Dynamical Core Developments at KIAPS 34/ 48 2D Global SWE Results Animation Cosine Bell Advection Deformational flow Rossby-Haurwitz T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 34/ 48

35 Dynamical Core Developments at KIAPS 35/ 48 2D Global SWE Results Test Case 1: Cosine Bell Advection T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 35/ 48

36 Dynamical Core Developments at KIAPS 36/ 48 2D Global SWE Results Deformational Flow T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 36/ 48

37 Dynamical Core Developments at KIAPS 37/ 48 2D Global SWE Results Test Case 2: Global Steady State Nonlinear Zonal Geostrophic Flow T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 37/ 48

38 Dynamical Core Developments at KIAPS 38/ 48 2D Global SWE Results Test Case 6: Rossby-Haurwitz Wave CG DG Day 0 Day 7 Day 15 T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 38/ 48

39 Dynamical Core Developments at KIAPS 39/ 48 Conclusion Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 39/ 48

40 Dynamical Core Developments at KIAPS 40/ 48 Conclusion CG/DG: Having High Order Numerics: Practical? For idealized advection, SWE test cases indicate that high order is actually more efficient Sensitive to IC (perfectly smooth IC required) Temporal discretization order must match Spatial order Presence of vertical physics forcing terms acting like point sources But having high order horizontal numerics is good Future-proof h/p refinement Better parallel scalability for higher spatial order T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 40/ 48

41 Dynamical Core Developments at KIAPS 41/ 48 Conclusion Summary KIAPS is aiming to develop Korea s next-generation operational model in 9 years Development strategy is based on unified CG/DG framework With CG/DG numerics, arbitrary spatial order can be achieved With m stage m 1 order SSP explicit RK time integrator, arbitrary temporal order can be achieved High order convergence properties of CG/DG on 1D Advection, SWE are tested CG/DG based global SWE with cartesian coordinate + Lagrange multiplier is tested No signs of cubed-sphere grid imprinting DG seems to be computationally more expensive compared to CG due to numerical flux computation Results indicate that CG/DG is robust T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 41/ 48

42 Dynamical Core Developments at KIAPS 42/ 48 Backup Slides Outline 1 Acknowledgments 2 KIAPS Introduction 3 KIAPS DyCore Design Considerations 4 1D CG/DG Advection, SWE Results 5 2D Global SWE Results 6 Conclusion 7 Backup Slides T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 42/ 48

43 Dynamical Core Developments at KIAPS 43/ 48 Backup Slides 2012 DCMIP Models Origin Eqns. Grid Structure Horizontal Vertical CAM-FV NCAR H lat-lon C D Lin-Rood CAM-SE NCAR H Cubed S SE sigma-p L Dynamico France H Icos(hex) C sigma-p L ENDGame UKMO N lat-lon C sigma-z CP FIM NOAA H Icos (hex) A theta FV3 GFDL H Cubed S C Lin-Rood GEM Env Ca H lat-lon C sigma-p CP GEM-YY Env Ca H Yin-Yang C sigma-p CP IFS ECMWF H/N Gaussian SH FEM ICON-IAP IAP N Icos (hex) C sigma-z L ICON MPI-DWD H/N Icos (tri) C H:s-p, N:s-z, L MCORE U Michigan N Cubed S A MPAS NCAR H/N SCVT C H:s-p, N:s-z, L NICAM Japan N Icos (hex) A NIM NOAA N Icos (hex) A sigma-z L OLAM U Miami H Icos (tri) cut-cell z PUMA U Hamburg H Gaussian SH sigma-p UZIM CSU N Icos (hex) Z Arakawa Konor T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 43/ 48

44 Dynamical Core Developments at KIAPS 44/ 48 Backup Slides Galerkin Method Galerkin method utilizes the weak integral form instead of the differential form for discretization q t + f ( x = 0 qn Φ i + f ) N dω e = 0 Ω e t x q N Φ i Ω e t dωe + q N (x, t) = Ω e N q j (t)φ j (x) j=0 x (Φ if N )dω e Ω e Φ i q N t dωe + [Φ if N ] Γ Ω e Ω e Φ i x f NdΩ e = 0 Φ i x f NdΩ e = 0 T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 44/ 48

45 Dynamical Core Developments at KIAPS 45/ 48 Backup Slides Galerkin Method (Cont.) q j Φ i Φ j Ω e t dωe + [Φ Φ i iφ j f j ] Γ Ω e x Φ jf j dω e = 0 M (e) ij Φ i Φ j dω e Ω e D (e) dφ i ij dx Φ jdω e M (e) ij dq dt (e) F (e) ij Ω e Φ i Φ j + F (e) ij f (e) j D ij f (e) j = 0 T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 45/ 48

46 Dynamical Core Developments at KIAPS 46/ 48 Backup Slides Continuous Galerkin (CG) DSS (Direct Stiffness Summation) N e M IJ e=1 M (e) ij, D N e IJ D (e) ij e=1 N e N e q J q j, f J e=1 For CG, basis functions must be C 0 continuous (via DSS) Thus, flux terms (jump conditions calculated at the element boundary Γ) disappear everywhere except for the model lateral boundary e=1 f j M IJ dq J dt D IJ f J = 0 dq J = M 1 IJ D IJ f J dt Global CG: Spectral, Pseudo-Spectral Local CG: Finite-Element, Spectral-Element T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 46/ 48

47 Dynamical Core Developments at KIAPS 47/ 48 Backup Slides Discontinuous Galerkin (DG) All calculations are element based because element-wise discontinuity is allowed f ( ) j : Numerical flux DG Family M (e) ij (e) dq dt dq dt (e) = M 1(e) ij + F (e) ij f ( ) j D ij f (e) j = 0 ( F (e) ij f ( ) j + D ij f (e) j ) SE (when DSS is applied) FV (when piecewise constant basis function is used) Spectral Multi-Domain Penalty Method (Strong form DG) Spectral Difference Method (Strong form DG) T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 47/ 48

48 Dynamical Core Developments at KIAPS 48/ 48 Bibliography I Côté, J., 1988: A lagrange multiplier approach for the metric terms of semi-lagrangian models on the sphere. Quart. J. Roy. Meteor. Soc., 114, Dennis, J., et al., 2012: Cam-se: A scalable spectral element dynamical core for the community atmosphere model. Int. J. High Perf. Comput. Appl., 74 89, doi: / Giraldo, F. X., 2001: A spectral element shallow water model on spherical geodesic grids. Int. J. Numer. Meth. Fluids, 35, Giraldo, F. X., 2012: Element-based Galerkin Methods. Naval Postgraduate School, 288 pp. Gottlieb, S., 2005: On high order strong stability preserving runge kutta and multi step time discretizations. J. Sci. Comput., 25 (1), Kelly, J. F. and F. X. Giraldo, 2012: Continuous and discontinuous galerkin methods for a scalable three-demensional nondydrostatic atmospheric model: Limited-area mode. J. Comput. Physics., 231, , doi: /j.jcp T.-J. Oh, S.-J. Choi, T.-H. Yi, S.-H. Kang, and Y.-J. Kim Dynamical Core Developments at KIAPS 48/ 48

The Nonhydrostatic Unified Model of the Atmosphere (NUMA): CG Dynamical Core

The Nonhydrostatic Unified Model of the Atmosphere (NUMA): CG Dynamical Core Frank Giraldo Department of Applied Mathematics Naval Postgraduate School Monterey CA 93943 http://faculty.nps.edu/projects/numa