Computational Math Group

Amik St-Cyr, Ram Nair, Natasha FlyerGroup Head: Piotr Smolarkiewicz

Group Goals and Research

Goals:

• Develop novel numerical methodologies for the geosciences• Bridge communities of applied/numerical math with geoscience

Background of the Group:

1. Applied mathematics

2. Computational mathematics

3. Geophysical fluid dynamics

4. Numerical weather predication

Examples of our research:• High-Order Method Modeling Environment (HOMME)• Non-conforming spectral element model • Radial Basis Functions (Meshless method)

Hydrostatic Dynamical Core (HOMME)

• Discontinuous Galerkin based new generation dynamical core development in HOMME framework (High-Order Method Modeling Environment)

• Inherently conservative, Geometric flexibility and highly scalable

[Simulated temperature field for the J-W baroclinic instability test at a resolution 0.7 degree with DG/HOMME]

Ongoing Research: Extend HOMME further to a full-fledge conservative dynamical core by incorporating NCAR-CAM physics packages.

Barotropic Vorticity EvolutionNon-conforming spectral element model on the sphere

0.3125 degrees...

RBFs: Moving Vortices on a Sphere (Flyer and Lehto ’09, Nair and Jablonowski ‘08)

Moving vortices on a sphere (RBFs)

12 Days Simulation, N = 3136, Time-Step= 20 minutes (RK4)

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Collaborations

NCAR: CGD, HAO, MMM

National: Courant Institute, Oakridge National Labs, Sandia National Labs, NCEP, Temple University, Columbia University, University of Michigan, University of Colorado-Boulder, University of Wyoming, North Carolina State, University of Minnesota, Florida State University, University of California-Davis, Boise State University, Arizona State University, Wichita State University

International: Chinese Academy of Science, UK Met office, University of Copenhagen (DK), Uppsala University (S), University of Cambridge (UK), University of Oxford (UK), Kyungpook National University (Korea), University of Stellenbosch (South Africa), University of Victoria (Canada), University of Geneva (CH), Universite Louvain la Neuve (B)

Community InteractionOrganized Workshops/Mini-symposiums• European Conference on Numerical Mathematics (ENUMATH)• PDEs on the Sphere • Korea SIAM Annual Meeting • SIAM Computational Issues in the Geosciences • SIAM Annual meeting• SIAM Parallel processing• SIAM Computational Science and Engineering• International Conference on Computational Science• International Conference on Spectral and High Order Methods• International Conference on Domain Decomposition Methods• Copper Mountain Conference on Iterative Methods• NCAR ASP Summer Colloquium 2008

Student/Post-doctoral Mentoring-SupportUniversity of Oklahoma, University of Colorado, University of Minnesota, North Carolina State, University of Michigan, University of Wyoming, Indian Institute of Science (IISc) Bangalore, Indian Institute of Technology (IIT) Madras, University of Toronto, Uppsala University (Sweden)

Geophysical Modeling Motivations for Research

Examples:

• Modeling coupling • Necessary scalability • Non-hydrostatic dynamics for realism• Free-boundary problems• Geometric flexibility• Algorithmic simplicity• Realistic time-stepping

Bottom LineHigh-resolution and numerical accuracy at low computational costs to resolve the multi-scale features of the earth system

Meeting the Computational Challenges

Advancing the frontier:

• High-order accurate methods– Meshless methods– Finite volume methods– Continuous and discontinuous Galerkin methods

• Scalable Numerics:– Conservative re-mapping of arbitrary grids– Optimized Schwarz solvers– Time-integrators (Lagrangian, Eulerian)

• Conservative non-oscillatory transport schemes• Adaptive mesh-refinement with error estimation• Unstructured meshes• Peta-Scale capable algorithms

Thank You