The Spectral Element Method for the Shallow Water Equations on the Sphere

Taylor, Mark A.

doi:10.1006/jcph.1996.5554

Cited by 282 publications

(265 citation statements)

References 10 publications

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“…On a cubed-sphere grid, a spherical quadrilateral grid, the following works [37,33,31,45,35] achieved a two-dimensional momentum representation avoiding any pole problem. Our new model achieves the same flexibility but on unstructured triangular grids using spherical triangular coordinates.…”

Section: Introductionmentioning

confidence: 99%

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

Läuter

Giraldo

Handorf

et al. 2008

Journal of Computational Physics

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a b s t r a c tA global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R 3 , are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a twodimensional representation of tangential momentum and therefore only two discrete momentum equations.The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step.The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211-224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of OðDx kþ1 Þ was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Dt is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

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Section: Introductionmentioning

confidence: 99%

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

Läuter

Giraldo

Handorf

et al. 2008

Journal of Computational Physics

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“…An alternative to doing this is the spectral-element method. The latter has been implemented on the sphere using a tiling of the sphere with quadrangular elements (Taylor, 1997;Baer et al, 2006). On such a tiling, accurate but not exact quadrature can be used.…”

Section: Discussionmentioning

confidence: 99%

“…The resulting method is spectrally accurate but not de-aliased. Therefore viscosity or filtering is required to remove small-scale noise (Taylor, 1997). Strictly speaking, integral invariants are not guaranteed to be conserved, but this may not be significant if sufficiently high-order elements are used.…”

Section: Discussionmentioning

confidence: 99%

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

Dubos

2009

Quart J Royal Meteoro Soc

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Solving transport equations on the whole sphere using an explicit time stepping and an Eulerian formulation on a latitude-longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost.The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourierfinite-element spatial discretization. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N 2 degrees of freedom, the computational cost is O (N 2 log N), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs O(N 2 ), and less than with a spherical harmonics method, which costs O(N 3 ). Differential operators with latitude-dependent coefficients are inverted at a cost of O(N 2 ).We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier-Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described.

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“…The HOMME atmospheric component model option of the CCSM uses a spectral element spatial discretization scheme, with a full description given elsewhere [11,12]. The cubed sphere uses a tiled, inscribed cube mapped to the sphere, which avoids the very disparate horizontal grid sizes near the poles that exist within a traditional latitude/longitude discretization.…”

Section: Jfnk Integration In Homme Shallow Water Test Casementioning

confidence: 99%

“…Although only h is solved here, the full nonlinear solution framework outlined in section 2 is fully implemented. The specific parameters are matched to earlier TC1 runs with HOMME using the fully explicit [11] and semi-implicit methods [12], whereby the anomaly is advected over the corners of the cubed sphere edges using a fixed zonal rotation at an angle π/4 from the Earth's equatorial axis and the spatial grid is set to 96x16x16, which corresponds to 24576 grid points and an average resolution of about 167 km (minimum 38 km and corresponding CFL limit of 36 s). The net time to solution for a 12 day simulation (throughout, day 1 is not included) of the explicit and JFNK method with a 30 s time step size on the Jaguar Cray XT4 on 96 processors is 4 min 53 s and 102 min 49 s with a final L 2 norm of error of 3 and 1.7 × 10 −3 respectively.…”

Section: Jfnk Integration In Homme Shallow Water Test Casementioning

confidence: 99%

A Scalable and Adaptable Solution Framework within Components of the Community Climate System Model

Evans

Rouson

Salinger

et al. 2009

Lecture Notes in Computer Science

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Abstract.A framework for a fully implicit solution method is implemented into (1) the High Order Methods Modeling Environment (HOMME), which is a spectral element dynamical core option in the Community Atmosphere Model (CAM), and (2) the Parallel Ocean Program (POP) model of the global ocean. Both of these models are components of the Community Climate System Model (CCSM). HOMME is a development version of CAM and provides a scalable alternative when run with an explicit time integrator. However, it suffers the typical time step size limit to maintain stability. POP uses a time-split semi-implicit time integrator that allows larger time steps but less accuracy when used with scale interacting physics. A fully implicit solution framework allows larger time step sizes and additional climate analysis capability such as model steady state and spin-up efficiency gains without a loss in scalability. This framework is implemented into HOMME and POP using a new Fortran interface to the Trilinos solver library, ForTrilinos, which leverages several new capabilities in the current Fortran standard to maximize robustness and speed. The ForTrilinos solution template was also designed for interchangeability; other solution methods and capability improvements can be more easily implemented into the models as they are developed without severely interacting with the code structure. The utility of this approach is illustrated with a test case for each of the climate component models.

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The Spectral Element Method for the Shallow Water Equations on the Sphere

Cited by 282 publications

References 10 publications

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

A Scalable and Adaptable Solution Framework within Components of the Community Climate System Model

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