Richard D. Loft

A Discontinuous Galerkin Global Shallow Water Model

Abstract A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

A Discontinuous Galerkin Transport Scheme on the Cubed Sphere

Abstract A conservative transport scheme based on the discontinuous Galerkin (DG) method has been developed for the cubed sphere. Two different central projection methods, equidistant and equiangular, are employed for mapping between the inscribed cube and the sphere. These mappings divide the spherical surface into six identical subdomains, and the resulting grid is free from singularities. Two standard advection tests, solid-body rotation and deformational flow, were performed to evaluate the DG scheme. Time integration relies on a third-order total variation diminishing (TVD) Runge–Kutta scheme without a limiter. The numerical solutions are accurate and neither exhibit shocks nor discontinuities at cube-face edges and vertices. The numerical results are either comparable or better than a standard spectral element method. In particular, it was found that the standard relative error metrics are significantly smaller for the equiangular as opposed to the equidistant projection.

Terascale Spectral Element Dynamical Core for Atmospheric General Circulation Models

Climate modeling is a grand challenge problem where scientific progress is measured not in terms of the largest problem that can be solved but by the highest achievable integration rate. These models have been notably absent in previous Gordon Bell competitions due to their inability to scale to large processor counts. A scalable and efficient spectral element atmospheric model is presented. A new semi-implicit time stepping scheme accelerates the integration rate relative to an explicit model by a factor of two, achieving 130 years per day at T63L30 equivalent resolution. Execution rates are reported for the standard shallow water and Held-Suarez climate benchmarks on IBM SP clusters. The explicit T170 equivalent multi-layer shallow water model sustains 343 Gflops at NERSC, 206 Gflops at NPACI (SDSC) and 127 Gflops at NCAR. An explicit Held-Suarez integration sustains 369 Gflops on 128 16-way IBM nodes at NERSC.

Semi-Implicit Spectral Element Atmospheric Model

A semi-implicit spectral element shallow water model on the cubed-sphere is described and numerical results are compared with an explicit formulation and the traditional spectral transform method using the standard test cases proposed by Williamson et al. (1992). The explicit time step is limited by the phase speed of the fastest gravity waves. Semi-implicit time integration schemes remove this stability restriction but require the solution of an elliptic problem. A weak variational formulation of the governing equations leads to a symmetric Helmholtz operator. The resulting implicit problem for the geopotential is then solved using a block-Jacobi preconditioned conjugate gradient solver. The simulation rate of the semi-implicit model is accelerated relative to the explicit model for practical climate resolutions by a factor of two.

The NCAR Spectral Element Climate Dynamical Core: Semi-Implicit Eulerian Formulation

AbstractA prototype dynamical core for the Community Atmospheric Model (CAM) component of the Community Climate System Model (CCSM) is presented. The 3D governing primitive equations are specified in curvilinear coordinates on the cubed-sphere combined with a hybrid pressure η vertical coordinate. The horizontal space discretisation is based on a

Early experience with scientific applications on the blue gene/l supercomputer

\mathbb{P}_N - \mathbb{P}_N

Architectural and application: the performance of the NEC SX-4 on the NCAR benchmark suite

spectral element variational formulation. A semi-implicit time integration scheme is derived in order to circumvent the severe time step restrictions associated with gravity waves. Eigen-mode decomposition of the vertical structure matrix results in a set of decoupled 2D modified Helmholtz problems which are solved using a preconditioned conjugate gradient iteration. An idealized climate simulation is presented, where the semi-implicit scheme permits a much larger time step

EARLY EXPERIENCES WITH THE 360TF IBM BLUE GENE/L PLATFORM

Abstract We present implementation and performance issues of a data parallel version of the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM2). We describe automatic conversion tools used to aid in converting a production code written for a traditional vector architecture to data parallel code suitable for the Thinking Machines Corporation CM-5. Also, we describe the 3-D transposition method used to parallelize the spherical harmonic transforms in CCM2. This method employs dynamic data mapping techniques to improve data locality and parallel efficiency of these computations. We present performance data for the 3-D transposition method on the CM-5 for machine size up to 512 processors. We conclude that the parallel performance of the 3-D transposition method is adversely affected on the CM-5 by short vector lengths and array padding. We also find that the CM-5 spherical harmonic transforms spend about 70% of their execution time in communication. We detail a transposition-based data parallel implementation of the semi-Lagrangian Transport (SLT) algorithm used in CCM2. We analyze two approaches to parallelizing the SLT, called the departure point and arrival point based methods. We develop a performance model for choosing between these methods. We present SLT performance data which shows that the localized horizontal interpolation in the SLT takes 70% of the time, while the data remapping itself only require approximately 16%. We discuss the importance of scalable I/O to CCM2, and present the I/O rates measured on the CM-5. We compare the performance of the data parallel version of CCM2 on a 32-processor CM-5 with the optimized vector code running on a single processor Cray Y-MP. We show that the CM-5 code is 75% faster. We also give the overall performance of CCM2 running at higher resolutions on different numbers of CM-5 processors. We conclude by discussing the significance of these results and their implications for data parallel climate models.