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40
A discontinuous Galerkin global shallow water model
 Monthly Weather Review
, 2005
"... 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 cons ..."
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Cited by 28 (2 self)
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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 thirdorder 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 finitevolume models. 1.
A discontinuous Galerkin transport scheme on the cubedsphere, Monthly Weather Review 133
, 2005
"... 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 surf ..."
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Cited by 26 (4 self)
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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, solidbody rotation and deformational flow, were performed to evaluate the DG scheme. Time integration relies on a thirdorder total variation diminishing (TVD) Runge–Kutta scheme without a limiter. The numerical solutions are accurate and neither exhibit shocks nor discontinuities at cubeface 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. 1.
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
 In preparation; http://www.amath. washington.edu/~rjl/pubs/circles
, 2005
"... Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational do ..."
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Cited by 22 (6 self)
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Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the highresolution wavepropagation algorithm implemented in clawpack can be effectively used to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitudelongitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reactiondiffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.
Highorder trianglebased discontinuous Galerkin methods for hyperbolic equations on a rotating sphere.
 J. Comput. Phys.,
, 2006
"... Abstract Highorder trianglebased discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses highorder Lagrange polynomials on the triangle using ..."
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Cited by 20 (3 self)
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Abstract Highorder trianglebased discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses highorder Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite elementtype area integrals are evaluated using order 2N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal trianglebased spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129150], is small, local and efficient to invert. Two types of finite volumetype flux integrals are studied: a set based on GaussLobatto quadrature points (order 2N À 1) and a set based on Gauss quadrature points (order 2N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2N integration being the most accurate of the four DG methods studied. The strong advection form with 2N integration performed extremely well even for flows with shock waves. The strong conservation form with 2N À 1 integration yielded results almost as good as those with 2N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix. Published by Elsevier Inc.
The Evolution of Dynamical Cores for Global Atmospheric Models
 JOURNAL OF THE METEOROLOGICAL SOCIETY OF JAPAN 85B
, 2007
"... The evolution of global atmospheric model dynamical cores from the first developments in the early 1960s to present day is reviewed. Numerical methods for atmospheric models are not straightforward because of the socalled pole problem. The early approaches include methods based on composite meshes, ..."
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Cited by 17 (0 self)
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The evolution of global atmospheric model dynamical cores from the first developments in the early 1960s to present day is reviewed. Numerical methods for atmospheric models are not straightforward because of the socalled pole problem. The early approaches include methods based on composite meshes, on quasihomogeneous grids such as spherical geodesic and cubed sphere, on reduced grids, and on a latitudelongitude grid with short time steps near the pole, none of which were entirely successful. This resulted in the dominance of the spectral transform method after it was introduced. SemiLagrangian semiimplicit methods were developed which yielded significant computational savings and became dominant in Numerical Weather Prediction. The need for improved physical properties in climate modeling led to developments in shape preserving and conservative methods. Today the numerical methods development community is extremely active with emphasis placed on methods with desirable physical properties, especially conservation and shape preservation, while retaining the accuracy and efficiency gained in the past. Much of the development is based on quasiuniform grids. Although the need for better physical properties is emphasized in this paper, another driving force is the need to develop schemes which are capable of running efficiently on computers with thousands of processors and distributed memory. Test cases for dynamical core evaluation are also briefly reviewed. These range from well defined deterministic tests to longer term statistical tests with both idealized forcing and complete parameterization packages but simple geometries. Finally some aspects of coupling dynamical cores to parameterization suites are discussed.
Highorder finite volume WENO schemes for the shallow water equations with dry states
, 2011
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Discontinuous Galerkin Spectral/hp Element Modelling of Dispersive Shallow Water Systems
"... Twodimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a twodimensional system is obtained which approximates the full threedimensional problem. Nevertheless, for mo ..."
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Cited by 7 (0 self)
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Twodimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a twodimensional system is obtained which approximates the full threedimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretisation in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretisation of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.
Development of a Scalable Global Discontinuous Galerkin Atmospheric Model,’’ 2006; see http:// www.csc.cs.colorado.edu/;tufo/pubs/tufo2005ijcse.pdf
"... Abstract: An efficient and scalable global discontinuous Galerkin atmospheric model (DGAM) on the sphere is developed. The continuous flux form of the nonlinear shallow water equations on the cubedsphere (in curvilinear coordinates) are developed. Spatial discretization is a nodal basis set of Lege ..."
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Cited by 5 (2 self)
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Abstract: An efficient and scalable global discontinuous Galerkin atmospheric model (DGAM) on the sphere is developed. The continuous flux form of the nonlinear shallow water equations on the cubedsphere (in curvilinear coordinates) are developed. Spatial discretization is a nodal basis set of Legendre polynomials. Fluxes along internal element interfaces are approximated by a LaxFriedrichs scheme. A thirdorder strong stability preserving RungeKutta scheme is applied for time integration. The standard shallow water test suite of Williamson et al. (1992) is used to validate the model. It is observed that the numerical solutions are accurate, the model conserves mass to machine precision, and there are no spurious oscillations in a test case where zonal flow impinges a mountain. The serial execution time of the highorder nodal DG scheme presented here is half that of the modal version DG scheme. Development time was substantially reduced by building the model in the High Order Method Modeling Environment (HOMME) developed at the National Center for Atmospheric Research (NCAR). Performance and scaling data for the steady state geostrophic flow problem Williamson et al. (1992) is presented. Sustained performance of 8 % of peak is observed on 2048 processor of a IBM Blue Gene/L supercomputer.