The effect of the formulation of nonlinear terms on aliasing errors in spectral methods

Blaisdell, Gregory A.; Spyropoulos, Evangelos T.; Qin, Jianhua

doi:10.1016/0168-9274(96)00005-0

Cited by 169 publications

(110 citation statements)

References 8 publications

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“…Depending on the choice of the volume flux it is possible for the flux differencing scheme to recover discretisations of alternative split forms of the PDE. The split form of a PDE, often found by averaging the advective and conservative form of the equations, is known to have stabilisation properties for non-linear PDE discretisations [25,26,27]. But, due to their form, it is often unclear if the approximation remains globally conservative in the sense of Lax-Wendroff.…”

Section: Can Be Expressed In a Similar Indicial Formmentioning

confidence: 99%

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Wintermeyer

Winters

Gassner

et al. 2017

Journal of Computational Physics

100

114

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We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

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Section: Can Be Expressed In a Similar Indicial Formmentioning

confidence: 99%

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Wintermeyer

Winters

Gassner

et al. 2017

Journal of Computational Physics

100

114

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“…The skew-symmetric form of the advection terms is found to reduce aliasing errors in spectral calculations, and although comparatively computationally expensive, is found to give good results especially when resolution is at a premium [2,21,41]. The notations DSM-L, DSM-M and DSM-P will be used to denote the use of the dynamic Smagorinsky model with, respectively, filtering in the Legendre or modal polynomial bases via DPT, and Lagrange-interpolant projection filtering.…”

Section: Implementation Of Dynamic Lesmentioning

confidence: 99%

Spectral element filtering techniques for large eddy simulation with dynamic estimation

Blackburn

Schmidt

2003

Journal of Computational Physics

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Spectral element methods have previously been successfully applied to direct numerical simulation of turbulent flows with moderate geometrical complexity and low to moderate Reynolds numbers. A natural extension of application is to large eddy simulation of turbulent flows, although there has been little published work in this area. One of the obstacles to such application is the ability to deal successfully with turbulence modelling in the presence of solid walls in arbitrary locations. An appropriate tool with which to tackle the problem is dynamic estimation of turbulence model parameters, but while this has been successfully applied to simulation of turbulent wall-bounded flows, typically in the context of spectral and finite volume methods, there have been no published applications with spectral element methods. Here, we describe approaches based on element-level spectral filtering, couple these with the dynamic procedure, and apply the techniques to large eddy simulation of a prototype wall-bounded turbulent flow, the plane channel, using a mixing length-based eddy viscosity subgrid-scale model. The methods outlined here may be carried over without modification to more complex geometries.

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“…See [4] for a comparison of splitting methods with different formulations of the energy equation. For a heuristic discussion on aliasing errors for split approximations, see [1].…”

Section: Skew-symmetric Splittingmentioning

confidence: 99%

On Skew-Symmetric Splitting and Entropy Conservation Schemes for the Euler Equations

Sjögreen

Yee

2010

Numerical Mathematics and Advanced Applications 2009

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The Tadmor type of entropy conservation formulation for the Euler equations and various skewsymmetric splittings of the inviscid flux derivatives are discussed. Numerical stability of high order central and Padé type (centered compact) spatial discretization is enhanced through the application of these formulations. Numerical test on a 2-D vortex convection problem indicates that the stability and accuracy of these formulations using the same high order central spatial discretization are similar for vortex travel up to a few periods. For two to three times longer time integrations, their corresponding stability and accuracy behaviors are very different. The goal of this work is to improve treatment of nonlinear instabilities and to minimize the use of numerical dissipation in numerical simulations of shock-free compressible turbulence and turbulence with shocks .

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The effect of the formulation of nonlinear terms on aliasing errors in spectral methods

Cited by 169 publications

References 8 publications

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Spectral element filtering techniques for large eddy simulation with dynamic estimation

On Skew-Symmetric Splitting and Entropy Conservation Schemes for the Euler Equations

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