Dynamically adaptive spectral-element simulations of 2D incompressible Navier–Stokes vortex decays

Fournier, A.; Rosenberg, Duane; Pouquet, A.

doi:10.1080/03091920802684665

Cited by 4 publications

(5 citation statements)

References 30 publications

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“…Following the same procedure as for the discrete divergence theorem, stepping from (29) to (30), we see that the GLL quadrature will be exact with respect to dx c for the (38) terms involving @/@x c . As in the step from (30) to (31), replacing these sums with the integrals and changing the sum over c to a sum over the triplets (23), (38) becomes…”

Section: The Discrete Stokes Theorem For An Elementmentioning

confidence: 99%

“…Hence the (29) summand is a polynomial in x c of degree at most 2d À 1, evaluated at its GLL nodal values. Because GLL quadrature in the x c direction is exact for such polynomials, we can replace this sum by the x c -integral (30) which is then evaluated in (31).…”

Section: The Discrete Divergence Theorem For An Elementmentioning

confidence: 99%

“…We now give a summary of the SEM, using the traditional finite-element inner-product formulation with globally defined continuous basis functions [30,31]. This formulation allows for a clearer illustration of the numerical properties of the method, while the more standard matrix-vector formulation [32,33] is useful for efficient numerical implementations.…”

Section: Spectral-element Discretizationmentioning

confidence: 99%

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A compatible and conservative spectral element method on unstructured grids

Taylor

Fournier

2010

Journal of Computational Physics

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193

155

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Section: The Discrete Stokes Theorem For An Elementmentioning

confidence: 99%

Section: The Discrete Divergence Theorem For An Elementmentioning

confidence: 99%

Section: Spectral-element Discretizationmentioning

confidence: 99%

See 1 more Smart Citation

A compatible and conservative spectral element method on unstructured grids

Taylor

Fournier

2010

Journal of Computational Physics

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193

155

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“…This means that Q T ji = h j (⇠ i ) has the property that all the columns (i = 1, ..., m) sum to 1. Let us call this the Summation Property of Q T (see also [22]). …”

Section: Definitions and Properties Of Matricesmentioning

confidence: 99%

“…Additional discussions regarding the DSS operation on non-conforming edges can be found in [8,6,7]. In [22] the authors introduce continuous global basis functions for non-conforming CG, which intrinsically maintain continuity in general situations in arbitrary dimensions.…”

Section: Overview Of the Cg And Dg Methodsmentioning

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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A Non-oscillatory Advection Operator for the Compatible Spectral Element Method

Taylor

St.-Cyr

Fournier

2009

Lecture Notes in Computer Science

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Abstract. The spectral element method is well known as an efficient way to obtain high-order numerical solutions on unstructured finite element grids. However, the oscillatory nature of the method's advection operator makes it unsuitable for many applications. One popular way to address this problem is with high-order discontinuous-Galerkin methods. In this work, an alternative solution which fits within the continuous Galerkin formulation of the spectral element method is proposed. Making use of a compatible formulation of spectral elements, a natural way to implement conservative non-oscillatory reconstructions for spectral element advection is shown. The reconstructions are local to the element and thus preserve the parallel efficiency of the method. Numerical results from a low-order quasi-monotone reconstruction and a higher-order signpreserving reconstruction are presented.

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Dynamically adaptive spectral-element simulations of 2D incompressible Navier–Stokes vortex decays

Cited by 4 publications

References 30 publications

A compatible and conservative spectral element method on unstructured grids

A compatible and conservative spectral element method on unstructured grids

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

A Non-oscillatory Advection Operator for the Compatible Spectral Element Method

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