Gassner, Gregor J. (2014). A kinetic energy preserving nodal discontinuous Galerkin spectral element method. Int. J. Numer. Methods Fluids, 76 (1). S. 28 - 51. HOBOKEN: WILEY-BLACKWELL. ISSN 1097-0363

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Abstract

In this work, we discuss the construction of a skew-symmetric discontinuous Galerkin (DG) collocation spectral element approximation for the compressible Euler equations. Starting from the skew-symmetric formulation of Morinishi, we mimic the continuous derivations on a discrete level to find a formulation for the conserved variables. In contrast to finite difference methods, DG formulations naturally have inter-domain surface flux contributions due to the discontinuous nature of the approximation space. Thus, throughout the derivations we accurately track the influence of the surface fluxes to arrive at a consistent formulation also for the surface terms. The resulting novel skew-symmetric method differs from the standard DG scheme by additional volume terms. Those volume terms have a special structure and basically represent the discretization error of the different product rules. We use the summation-by-parts (SBP) property of the Gauss-Lobatto-based DG operator and show that the novel formulation is exactly conservative for the mass, momentum, and energy. Finally, an analysis of the kinetic energy balance of the standard DG discretization shows that because of aliasing errors, a nonzero transport source term in the evolution of the discrete kinetic energy mean value may lead to an inconsistent increase or decrease in contrast to the skew-symmetric formulation. Furthermore, we derive a suitable interface flux that guarantees kinetic energy preservation in combination with the skew-symmetric DG formulation. As all derivations require only the SBP property of the Gauss-Lobatto-based DG collocation spectral element method operator and that the mass matrix is diagonal, all results for the surface terms can be directly applied in the context of multi-domain diagonal norm SBP finite difference methods. Numerical experiments are conducted to demonstrate the theoretical findings. Copyright (C) 2014 John Wiley & Sons, Ltd.

Item Type: Journal Article
Creators:
CreatorsEmailORCIDORCID Put Code
Gassner, Gregor J.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-428830
DOI: 10.1002/fld.3923
Journal or Publication Title: Int. J. Numer. Methods Fluids
Volume: 76
Number: 1
Page Range: S. 28 - 51
Date: 2014
Publisher: WILEY-BLACKWELL
Place of Publication: HOBOKEN
ISSN: 1097-0363
Language: English
Faculty: Unspecified
Divisions: Unspecified
Subjects: no entry
Uncontrolled Keywords:
KeywordsLanguage
NAVIER-STOKES EQUATIONS; FINITE-DIFFERENCE SCHEMES; LARGE-EDDY SIMULATION; ELLIPTIC PROBLEMS; ORDER; SUMMATION; PARTS; FLOW; STABILITY; BOUNDARYMultiple languages
Computer Science, Interdisciplinary Applications; Mathematics, Interdisciplinary Applications; Mechanics; Physics, Fluids & PlasmasMultiple languages
URI: http://kups.ub.uni-koeln.de/id/eprint/42883

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