Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations

Kopera, Michal A.; Giraldo, Francis X.

doi:10.1016/j.jcp.2014.06.026

Cited by 74 publications

(62 citation statements)

References 45 publications

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“…This means that both methods use the same data structures and routines for integration, differentiation etc., which allows for a direct comparison of the key differences. We described the DG approach in detail in a previous publication (see [1]). Here we want to highlight the CG method and the details of its implementation in NUMA.…”

Section: Overview Of the Cg And Dg Methodsmentioning

confidence: 99%

“…A detailed definition of those matrices can be found in [1]. At this point it is important to mention that the mass matrix can be constructed such that it is diagonal (by using inexact integration, see [16]), and therefore trivial to invert.…”

Section: Overview Of the Cg And Dg Methodsmentioning

confidence: 99%

“…We use the conservative form of the Euler equations described in detail in [16,1]. Here we provide the equation set for completeness.…”

Section: Governing Equationsmentioning

confidence: 99%

“…As explained in [1], simple sequential summation of mass contributions from all the nodal points can result in a considerable roundoff error, which blurs the outcome of the mass conservation study. In order to control the roundoff error in the mass computation, we use a pairwise summation algorithm outlined in [23] 2 .…”

Section: Computation Of the Massmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Overview Of the Cg And Dg Methodsmentioning

confidence: 99%

Section: Overview Of the Cg And Dg Methodsmentioning

confidence: 99%

“…We use the conservative form of the Euler equations described in detail in [16,1]. Here we provide the equation set for completeness.…”

Section: Governing Equationsmentioning

confidence: 99%

Section: Computation Of the Massmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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A Godunov‐type finite‐volume solver for nonhydrostatic Euler equations with a time‐splitting approach

Nazari

Nair

2017

J Adv Model Earth Syst

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A two‐dimensional conservative nonhydrostatic (NH) model based on the compressible Euler system has been developed in the Cartesian (x, z) domain. The spatial discretization is based on a Godunov‐type finite‐volume (FV) method employing dimensionally split fifth‐order reconstructions. The model uses the explicit strong stability‐preserving Runge‐Kutta scheme and a split‐explicit method. The time‐split approach is generally based on the split‐explicit method, where the acoustic modes in the Euler system are solved using small time steps, and the advective modes are treated with larger time steps. However, for the Godunov‐type FV method this traditional approach is not trivial for the Euler system of equations. In the present study, a new strategy is proposed by which the Euler system is split into three modes, and a multirate time integration is performed. The computational efficiency of the split scheme is compared with the explicit one using the FV model with various NH benchmark test cases.

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Simulation of shallow‐water jets with a unified element‐based continuous/discontinuous Galerkin model with grid flexibility on the sphere

Marras

Kopera

Giraldo

2014

Quart J Royal Meteoro Soc

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We test the behaviour of a unified continuous/discontinuous Galerkin (CG/DG) shallowwater model in spherical geometry with curved elements on three different grids of ubiquitous use in atmospheric modelling: (i) the cubed-sphere, (ii) the reduced latitude-longitude, and (iii) the icosahedral grid. Both conforming and non-conforming grids are adopted including static and dynamically adaptive grids for a total of twelve mesh configurations. The behaviour of CG and DG on the different grids are compared for a nonlinear midlatitude perturbed jet and for a linear case that admits an analytic solution. Because the inviscid solution on certain grids shows a high sensitivity to the resolution, the viscous counterpart of the governing equations is also solved and the results compared. The logically unstructured element-based CG/DG model described in this article is flexible with respect to arbitrary grids. However, we were unable to define a best grid configuration that could possibly minimize the error regardless of the characteristic geometry of the flow. This is especially true if the governing equations are not regularized by the addition of a sufficiently large, fully artificial, diffusion mechanism, as will be shown. The main novelty of this study lies in the unified implementation of two element-based Galerkin methods that share the same numerical machinery and do not rely on any specific grid configuration to solve the shallow-water equation on the sphere.

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Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations

Cited by 74 publications

References 45 publications

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

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

A Godunov‐type finite‐volume solver for nonhydrostatic Euler equations with a time‐splitting approach

Simulation of shallow‐water jets with a unified element‐based continuous/discontinuous Galerkin model with grid flexibility on the sphere

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