Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode

Kelly, James Floyd; Giraldo, Francis X.

doi:10.1016/j.jcp.2012.04.042

Cited by 108 publications

(111 citation statements)

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“…This permits the computation of volume integrals with communication of interprocessor boundary data, thereby, giving DG an edge in-terms of parallel performance. Indeed this was the main reason for the better performance of DG compared to CG for the results reported in [21]. We should note that CG can benefit from a similar splitting of the volume integrals into two parts: integration for interior elements and that for elements that have at least one shared inter-processor face.…”

Section: Algorithm 1 Indexing Function For Cgmentioning

confidence: 89%

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Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Abdi

Giraldo

2016

Journal of Computational Physics

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A unified approach for the numerical solution of the 3D hyperbolic Euler equations using high order methods, namely continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, is presented. First, we examine how classical CG that uses a global storage scheme can be constructed within the DG framework using constraint imposition techniques commonly used in the finite element literature. Then, we implement and test a simplified version in the Non-hydrostatic Unified Model of the Atmosphere (NUMA) for the case of explicit time integration and a diagonal mass matrix. Constructing CG within the DG framework allows CG to benefit from the desirable properties of DG such as, easier hp-refinement, better stability etc. Moreover, this representation allows for regional mixing of CG and DG depending on the flow regime in an area. The different flavors of CG and DG in the unified implementation are then tested for accuracy and performance using a suite of benchmark problems representative of cloud-resolving scale, meso-scale and global-scale atmospheric dynamics. The value of our unified approach is that we are able to show how to carry both CG and DG methods within the same code and also offer a simple recipe for modifying an existing CG code to DG and vice versa.

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Section: Algorithm 1 Indexing Function For Cgmentioning

confidence: 89%

“…3. The details of the parallel implementations of CG and DG in NUMA can be found in [21]. DG benefits from a communication-computation overlap while computing fluxes.…”

Section: Algorithm 1 Indexing Function For Cgmentioning

confidence: 99%

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Abdi

Giraldo

2016

Journal of Computational Physics

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“…2 we outline the governing equations and give an overview of the unified CG/DG method implemented in the Non-hydrostatic Unified Model of the Atmosphere (NUMA) [14,15]. We discuss the data structures and routines shared by both methods and point out the differences.…”

Section: Introductionmentioning

confidence: 99%

Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

Kopera

Giraldo

2015

Journal of Computational Physics

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We perform a comparison of mass conservation properties of the continuous (CG) and discontinuous (DG) Galerkin methods on non-conforming, dynamically adaptive meshes for two atmospheric test cases. The two methods are implemented in a unified way which allows for a direct comparison of the non-conforming edge treatment. We outline the implementation details of the non-conforming direct stiffness summation algorithm for the CG method and show that the mass conservation error is similar to the DG method. Both methods conserve to machine precision, regardless of the presence of the non-conforming edges. For lower order polynomials the CG method requires additional stabilization to run for very long simulation times. We addressed this issue by using filters and/or additional artificial viscosity. The mathematical proof of mass conservation for CG with non-conforming meshes is presented in Appendix B.Keywords: adaptive mesh refinement, continuous Galerkin method, discontinuous Galerkin method, non-conforming mesh, compressible Euler equations, atmospheric simulations IntroductionBoth element-based Galerkin methods and adaptive mesh refinement for atmospheric modeling are active fields of research. In [1] we provide an overview of the literature covering both topics, with particular attention to the discontinuous Galerkin method. In this paper we extend that work by comparing the continuous (CG) and discontinuous Galerkin (DG) methods with non-conforming adaptive mesh refinement. As a metric for the comparison we chose the mass conservation, as it is an important feature for many atmospheric applications. We believe this is a good metric because, for smooth solutions, both methods will yield similar accuracy; it is unclear how both methods will compare for non-smooth solutions but it is expected that they can achieve similarly good results, especially with the inclusion of stabilization techniques for both methods. However, this topic is beyond the scope of this paper but is something to address in the future. Both methods are implemented in a unified way, such that they use the same data structures and routines for computations, differing only in the inter-element communication, which is described in detail in Sec. 2.2. This approach allows us to be confident that the differences we see in the results are due to the different methods rather than their implementations.There is a vast collection of work in the literature on CG with geometrically non-conforming mesh refinement, starting with the early work of [2,3] and later [4,5]. Those approaches were based on the mortar element method, which employs integral projection to ensure a weak continuity across the non-conforming edges. In this paper we follow the approach by Fischer et al. [6], that uses an interpolation based method for reconciling the continuity condition on non-conforming edges. The two methods were compared in [7].

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“…SEM is local in nature because it has a large on-processor operation count (Kelly and Giraldo, 2012). SEM achieves this high level of scalability by decomposing the physical domain into smaller pieces with a small communication stencil.…”

Section: Introductionmentioning

confidence: 99%

“…This is the case because a 2-D slice model can effectively test the practical issues resulting from the vertical discretization and time integration prior to construction of a full 3-D model. Although we could discretize the vertical direction using SEM (as proposed in Giraldo, 2012, andGiraldo et al, 2013), we chose to use a finite difference method for discretization in the vertical direction because it provides an easy way to couple the dynamics and existing physics packages.…”

Section: Introductionmentioning

confidence: 99%

Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects

et al. 2014

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Abstract. The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge-Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia-gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied.The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.

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Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode

Cited by 108 publications

References 39 publications

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects

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