Dealiasing techniques for high-order spectral element methods on regular and irregular grids

Mengaldo, Gianmarco; Grazia, Daniele De; Moxey, David; Vincent, Peter; Sherwin, Spencer J.

doi:10.1016/j.jcp.2015.06.032

Cited by 106 publications

(86 citation statements)

References 29 publications

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“…in the DG method [11,23,24,29,31,35,43] and in the finite element method [5,8,16,20,26,27]. This paper exploits the version the discontinuous Galerkin method where the continuity and boundary conditions are enforced by the finite difference rule.…”

Section: Introductionmentioning

confidence: 99%

Very high order discontinuous Galerkin method in elliptic problems

Jaśkowiec

2017

Comput Mech

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The paper deals with high-order discontinuous Galerkin (DG) method with the approximation order that exceeds 20 and reaches 100 and even 1000 with respect to one-dimensional case. To achieve such a high order solution, the DG method with finite difference method has to be applied. The basis functions of this method are high-order orthogonal Legendre or Chebyshev polynomials. These polynomials are defined in one-dimensional space (1D), but they can be easily adapted to two-dimensional space (2D) by cross products. There are no nodes in the elements and the degrees of freedom are coefficients of linear combination of basis functions. In this sort of analysis the reference elements are needed, so the transformations of the reference element into the real one are needed as well as the transformations connected with the mesh skeleton. Due to orthogonality of the basis functions, the obtained matrices are sparse even for finite elements with more than thousands degrees of freedom. In consequence, the truncation errors are limited and very high-order analysis can be performed. The paper is illustrated with a set of benchmark examples of 1D and 2D for the elliptic problems. The example presents the great effectiveness of the method that can shorten the length of calculation over hundreds times.

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Section: Introductionmentioning

confidence: 99%

Very high order discontinuous Galerkin method in elliptic problems

Jaśkowiec

2017

Comput Mech

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“…Specifically, we found that DG (Q > P) and FR DG(Q > P) are identical when using Q > P for linear and nonlinear flux functions as well as for regular and irregular tensor-product meshes 3 . This indicates that polynomial aliasing sources for these two schemes are identical and can therefore be addressed using equivalent strategies, such as the consistent integration (through additional quadrature points) of the nonlinearities arising either from the equations themselves or from the geometry [11,12].…”

Section: Introductionmentioning

confidence: 99%

On the Connections Between Discontinuous Galerkin and Flux Reconstruction Schemes: Extension to Curvilinear Meshes

et al. 2015

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This paper investigates the connections between many popular variants of the well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-product meshes. We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over-or consistent-integration-based dealiasing methods. The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over-or consistent-integration in an equivalent manner for both the approaches.

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“…Since this case uses three-dimensional curvilinear elements, the choice of quadrature order can affect stability as shown in [26]. We therefore increase the number of integration points by a factor of 2, compared to the previous two-dimensional simulations, in each direction in order to avoid aliasing effects.…”

Section: Figurementioning

confidence: 99%

A p-adaptation method for compressible flow problems using a goal-based error indicator

Ekelschot

Moxey

Sherwin

et al. 2017

Computers & Structures

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An accurate calculation of aerodynamic force coefficients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier-Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coefficients. To reduce error in the calculated force coefficients whilst keeping computational cost minimal, we propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator as follows.Given an objective functional such as the aerodynamic force coefficients, we use control theory to derive an adjoint problem which provides the sensitivity of the functional with respect to changes in the flow variables, and assume that these changes are represented by the local truncation error. In its final form, the goal-based error indicator represents the effect of truncation error on the objective functional, suitably weighted by the adjoint solution. Both flow governing and adjoint equations are solved by the same high-order method, where we allow the degree of the polynomial within an element to vary across the mesh.We initially calculate a steady-state solution to the governing equations using resolve the force coefficients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.

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Dealiasing techniques for high-order spectral element methods on regular and irregular grids

Abstract: 14.07.15 KB. Accepted for pub, indefinite embargo applied until published then 12 month embarg

Cited by 106 publications

References 29 publications

Very high order discontinuous Galerkin method in elliptic problems

Very high order discontinuous Galerkin method in elliptic problems

On the Connections Between Discontinuous Galerkin and Flux Reconstruction Schemes: Extension to Curvilinear Meshes

A p-adaptation method for compressible flow problems using a goal-based error indicator

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