Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling

Yu, Meilin; Giraldo, Francis X.; Peng, Melinda S.; Wang, Zhi Jian

doi:10.1175/mwr-d-15-0134.1

Cited by 30 publications

(30 citation statements)

References 66 publications

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“…In some instances, excessive anisotropic diffusion provided by certain slope limiters can actually deteriorate the shape of the rising thermal, and the WENO-based limiter has always been found to yield the best overall results despite the fact that it could not completely suppress the Q 0 overshoots. The position of the tip of the rising bubble is, however, seen to converge at high resolution toward ;950 m. The Q 0 extreme values shown in Table 1 at all three resolutions appear to be very similar to those reported by Yu et al (2015) on a very similar configuration with loworder polynomials and selective artificial viscosity. Yu et al (2015) show that keeping the perturbation potential temperature signal within its expected bounds requires the use of high-resolution grids (Dx # 5 m) and at least eighth-order polynomials.…”

Section: ) Resultssupporting

confidence: 79%

“…The position of the tip of the rising bubble is, however, seen to converge at high resolution toward ;950 m. The Q 0 extreme values shown in Table 1 at all three resolutions appear to be very similar to those reported by Yu et al (2015) on a very similar configuration with loworder polynomials and selective artificial viscosity. Yu et al (2015) show that keeping the perturbation potential temperature signal within its expected bounds requires the use of high-resolution grids (Dx # 5 m) and at least eighth-order polynomials. Increasing the level of numerical diffusion may also help convergence, but this would be done at the expense of the overall quality of the results [see Yelash et al (2014) and discussion above].…”

Section: ) Resultssupporting

confidence: 79%

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Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

Savre

Percival

Herzog

et al. 2016

Monthly Weather Review

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This paper presents the first attempt to apply the compressible nonhydrostatic Active Tracer High-Resolution Atmospheric Model-Fluidity (ATHAM-Fluidity) solver to a series of idealized atmospheric test cases. ATHAMFluidity uses a hybrid finite-element discretization where pressure is solved on a continuous second-order grid while momentum and scalars are computed on a first-order discontinuous grid (also known as P 1DG 2 P 2 ). ATHAMFluidity operates on two-and three-dimensional unstructured meshes, using triangular or tetrahedral elements, respectively, with the possibility to employ an anisotropic mesh optimization algorithm for automatic grid refinement and coarsening during run time. The solver is evaluated using two-dimensional-only dry idealized test cases covering a wide range of atmospheric applications. The first three cases, representative of atmospheric convection, reveal the ability of ATHAM-Fluidity to accurately simulate the evolution of large-scale flow features in neutral atmospheres at rest. Grid convergence without adaptivity as well as the performances of the Hermite-Weighted Essentially Nonoscillatory (Hermite-WENO) slope limiter are discussed. These cases are also used to test the grid optimization algorithm implemented in ATHAM-Fluidity. Adaptivity can result in up to a sixfold decrease in computational time and a fivefold decrease in total element number for the same finest resolution. However, substantial discrepancies are found between the uniform and adapted grid results, thus suggesting the necessity to improve the reliability of the approach. In the last three cases, corresponding to atmospheric gravity waves with and without orography, the model ability to capture the amplitude and propagation of weak stationary waves is demonstrated. This work constitutes the first step toward the development of a new comprehensive limited area atmospheric model.

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Section: ) Resultssupporting

confidence: 79%

Section: ) Resultssupporting

confidence: 79%

Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

Savre

Percival

Herzog

et al. 2016

Monthly Weather Review

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“…In [20] the stability of the solution was achieved via a non-dissipative low pass spatial filter [52; 6]. For direct comparison, a viscous solution of the same problem is reported in [53] using a parameter-dependent element-based viscosity.…”

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confidence: 99%

Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES

Marras

Nazarov

Giraldo

2015

Journal of Computational Physics

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The high order spectral element approximation of the Euler equations is stabilized via a dynamic sub-grid scale model (Dyn-SGS). This model was originally designed for linear finite elements to solve compressible flows at large Mach numbers. We extend its application to high-order spectral elements to solve the Euler equations of low Mach number stratified flows. The major justification of this work is twofold: stabilization and large eddy simulation are achieved via one scheme only.Because the di usion coe cients of the regularization stresses obtained via Dyn-SGS are residual-based, the e ect of the artificial di usion is minimal in the regions where the solution is smooth. The direct consequence is that the nominal convergence rate of the high-order solution of smooth problems is not degraded. To our knowledge, this is the first application in atmospheric modeling of a spectral element model stabilized by an eddy viscosity scheme that, by construction, may fulfill stabilization requirements, can model turbulence via LES, and is completely free of a user-tunable parameter.From its derivation, it will be immediately clear that Dyn-SGS is independent of the numerical method; it could be implemented in a discontinuous Galerkin, finite volume, or other environments alike. Preliminary discontinuous Galerkin results are reported as well. The straightforward extension to non-linear scalar problems is also described. A suite of 1D, 2D, and 3D test cases is used to assess the method, with some comparison against the results obtained with the most known Lilly-Smagorinsky SGS model.

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“…The artificial viscosity µ is varied among the test cases. Note that NUMA is equipped to use either the local adaptive viscosity described in [17], or the dynamic sub-grid scale model for LES (Dyn-SGS) reported in [18]. However, to allow others to replicate our results, in this work we use a constant µ.…”

Section: Governing Equationsmentioning

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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Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling

Cited by 30 publications

References 66 publications

Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES

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

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