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Journal of Scientific Computing

Erschienen in:

10.03.2018

An Energy Stable Approach for Discretizing Hyperbolic Equations with Nonconforming Discontinuous Galerkin Methods

verfasst von: Jeremy E. Kozdon, Lucas C. Wilcox

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2018

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Abstract

When nonconforming discontinuous Galerkin methods are implemented for hyperbolic equations using quadrature, exponential energy growth can result even when the underlying scheme with exact integration does not support such growth. Using linear elasticity as a model problem, we propose a skew-symmetric formulation that has the same energy stability properties for both exact and inexact quadrature-based integration. These stability properties are maintained even when the material properties are variable and discontinuous, and the elements are non-affine (e.g., curved). Additionally, we show how the nonconforming scheme can be made conservative and constant preserving with variable material properties and curved elements. The analytic stability, conservation, and constant preserving results are confirmed through numerical experiments demonstrating the stability as well as the accuracy of the method.

Vorheriger Artikel A Globally and Quadratically Convergent Algorithm for Solving Multilinear Systems with -tensors

Nächster Artikel High Accurate Finite Differences Based on RBF Interpolation and its Application in Solving Differential Equations

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Even in the conforming case, the choice of the quadrature rule for some element types may require a skew-symmetric form even for constant coefficient problems on affine meshes (e.g., spectral element method (SEM) quadrature for the quadrilateral face of pyramids) [5].

In principle one could introduce approximations of the reference elements as well so that \({\hat{e}} \ne {\hat{e}}_{h}\), but since many methods are specified using straight-sided reference elements this is not considered here.

Nonconforming edge values need to be modified in order to ensure that new gaps and overlaps are not introduced along the edges of conforming elements.

In three-dimensions interpolation across nonconforming edges is necessary so that gaps and overlaps do not occur between conforming faces connected to nonconforming edges.

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Titel: An Energy Stable Approach for Discretizing Hyperbolic Equations with Nonconforming Discontinuous Galerkin Methods
verfasst von: Jeremy E. Kozdon
Lucas C. Wilcox
Publikationsdatum: 10.03.2018
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0682-1

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