Approximation error of the Lagrange reconstructing polynomial

Gerolymos, G. A.

doi:10.1016/j.jat.2010.09.007

Cited by 5 publications

(21 citation statements)

References 45 publications

(213 reference statements)

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“…The terms in ε ij -transport (1b) contain correlations of 1-order-higher derivatives of fluctuating quantities compared to the corresponding terms in r ij -transport (1a). Therefore, terms in the ε ij -transport equations (1b) are more sensitive to computational truncation errors (Gerolymos, 2011), requiring finer grids to achieve the same accuracy as the corresponding terms in the r ij -transport equations (1a). Furthermore, scaling analysis (Tennekes and Lumley, 1972, pp.…”

Section: Dns Computationsmentioning

confidence: 99%

Further analysis of the budgets of the dissipation tensor ${\varepsilon }_{{ij}}$ in turbulent plane channel flow

Gerolymos¹,

Vallet²

2017

Fluid Dyn. Res.

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Abstract. Recent DNS results [Gerolymos G.A., Vallet I. : J. Fluid Mech. 807 (2016) have provided data for the terms in the transport equations for the components of the dissipation tensor ε ij in low-Reynolds turbulent plane channel flow. The present paper extends the previous results by a detailed analysis of the behaviour of various mechanisms in the ε ij -transport equations (production, diffusion, redistribution, destruction), with particular emphasis on the component-by-component comparison with the corresponding mechanisms in the transport equations for the Reynolds-stresses r ij . The splitting of the pressure terms for the wall-normal components into redistribution and pressure-diffusion reveals substantially different behaviour near the wall. The wall-asymptotics of different terms in the transport equations are studied in detail, and examined using the DNS data. Both DNS data and wall-asympotic analysis show that the anisotropy of the destruction-of-dissipation tensor εε ij is fundamentally different from that of r ij or ε ij , never approaching the 2-component (2-C) state at the solid wall.

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Section: Dns Computationsmentioning

confidence: 99%

Further analysis of the budgets of the dissipation tensor ${\varepsilon }_{{ij}}$ in turbulent plane channel flow

Gerolymos¹,

Vallet²

2017

Fluid Dyn. Res.

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“…In a recent work [7] we have studied the exact and approximate reconstruction of a function h(x). We have obtained the general analytical solution of the deconvolution of Taylor-series problem [16, (3.13), pp.…”

Section: Reconstruction Backgroundmentioning

confidence: 99%

“…We have obtained the general analytical solution of the deconvolution of Taylor-series problem [16, (3.13), pp. 244-254], and used this solution in developing analytical relations for the approximation error of polynomial reconstruction on an arbitrary stencil in a homogeneous grid [7]. We briefly summarize those results of [7] which are the starting point of the analysis presented in the present work, and which are necessary for completeness.…”

Section: Reconstruction Backgroundmentioning

confidence: 99%

“…Alternatively, since the weights combine the interpolating (or reconstructing) polynomials on the substencils to exactly the interpolating (or reconstructing) polynomial on the entire stencil, they recover the highest possible accuracy (between weighted combinations of the substencils) and, for this reason, they are alternatively called optimal (in the sense of accuracy) weights [2,9]. The underlying linear interpolation or reconstruction used in WENO [5,3] schemes on the general stencil {i − M − , · · · , i + M + } (Definition 1.1) can be obtained by writing the approximation error [7, (56a), p. 292] for the K s + 1 (k s ∈ {0, · · · , K s }) substencils {i − M − + k s , · · · , i + M + − K s + k s } (Definition 1.1), each of which has an error of O(∆x M−K s +1 ) [7,Proposition 4.7,p. 292].…”

Section: Introductionmentioning

confidence: 99%

“…In §2 we very briefly summarize those results for the Lagrange reconstructing polynomial and its approximation error obtained in [7] which are necessary in the present work. 6 In §3 we study the basis polynomials In §4 we use the results of §3 to establish a 1-level subdivision rule (Lemma 4.2), by which, applying [12, Lemma 2.1], we construct (Proposition 4.5) an analytical recursive expression of the weight-functions for a general subdivision of an arbitrarily biased stencil on a homogeneous grid.…”

Section: Introductionmentioning

confidence: 99%

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Representation of the Lagrange reconstructing polynomial by combination of substencils

Gerolymos¹

2012

Journal of Computational and Applied Mathematics

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The Lagrange reconstructing polynomial [Shu C.W.: SIAM Rev. 51 (2009) 82-126] of a function f (x) on a given set of equidistant (∆x = const) points x i + ℓ∆x; ℓ ∈ {−M − , · · · , +M + } is defined as the polynomial whose sliding (with x) averages on [x − 1 2 ∆x, x + 1 2 ∆x] are equal to the Lagrange interpolating polynomial of f (x) on the same stencil [Gerolymos G.A.: J. Approx. Theory 163 (2011) 267-305]. We first study the fundamental functions of Lagrange reconstruction, show that these polynomials have only real and distinct roots, which are never located at the cellinterfaces (half-points) x i + n 1 2 ∆x (n ∈ Z), and obtain several identities. Using these identities, we show that there exists a unique representation of the Lagrange reconstructing polynomial on {i − M − , · · · , i + M + } as a combination of the Lagrange reconstructing polynomials on Neville substencils [Carlini E., Ferretti R., Russo G.: SIAM J. Sci. Comp. 27 (2005) 1071-1091], with weights which are rational functions of ξ (x = x i + ξ∆x) [Liu Y.Y., Shu C.W., Zhang M.P.: Acta Math. Appl. Sinica 25 (2009) 503-538], and give an analytical recursive expression of the weight-functions.We show that all of the poles of the rational weight-functions are real, and that there can be no poles at half-points. We then use the analytical expression of the weight-functions, combined with the factorization of the fundamental functions of Lagrange reconstruction, to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of ξ = 1 2 , iff all of the substencils contain either point i or point i + 1 (or both).

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A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation

Gerolymos

2012

Applied Mathematics and Computation

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In several applications, such as WENO interpolation and reconstruction [Shu C.W.: SIAM Rev. 51 (2009) , we are interested in the analytical expression of the weight-functions which allow the representation of the approximating function on a given stencil (Chebyshev-system) as the weighted combination of the corresponding approximating functions on substencils (Chebyshev-subsystems). We show that the weight-functions in such representations [Mühlbach G.: Num. Math. 31 (1978) 97-110] can be generated by a general recurrence relation based on the existence of a 1-level subdivision rule. As an example of application we apply this recurrence to the computation of the weight-functions for Lagrange interpolation [Carlini E., Ferretti R., Russo G.: SIAM J. Sci. Comp. 27 (2005) 1071-1091] for a general subdivision of the stencil {xcontaining the same number of M − K s + 1 points but each shifted by 1 cell with respect to its neighbour, and give a general proof for the conditions of positivity of the weight-functions (implying convexity of the combination), extending previous results obtained for particular stencils and subdvisions [Liu Y.Y., Shu C.W., Zhang M.P.: Acta Math. Appl. Sinica 25 (2009) 503-538]. Finally, we apply the recurrence relation to the representation by combination of substencils of derivatives of arbitrary order of the Lagrange interpolating polynomial.

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Approximation error of the Lagrange reconstructing polynomial

Cited by 5 publications

References 45 publications

Further analysis of the budgets of the dissipation tensor ${\varepsilon }_{{ij}}$ in turbulent plane channel flow

Further analysis of the budgets of the dissipation tensor ${\varepsilon }_{{ij}}$ in turbulent plane channel flow

Representation of the Lagrange reconstructing polynomial by combination of substencils

A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation

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