Central WENO Schemes Through a Global Average Weight

Baeza, Antonio; Bürger, Raimund; Mulet, Pep; Zorío, David

doi:10.1007/s10915-018-0773-z

Cited by 10 publications

(25 citation statements)

References 17 publications

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“…) [1,6,23,32,44]. Results about the parameters and the accuracy of , and class reconstructions in various finite volume settings are proven in [13,15,33].…”

Section: Introductionmentioning

confidence: 94%

“…When a discontinuity is present in the stencil, the nonlinear weights should deviate from their optimal values in order to avoid the occurrence of spurius oscillations in the numerical scheme. In practice, the nonlinear weights are computed with the help of oscillation indicators associated to each polynomial, that should be o (1) when the polynomial interpolates smooth data and O(1) when the polynomial interpolates discontinuous data. The construction is independent from the specific form of these indicators, which here we denote generically as OSC[P] ; typically the Jiang-Shu indicators from [21] are employed.…”

Section: The Novel Cwenozb Reconstruction In One Space Dimensionmentioning

confidence: 99%

See 1 more Smart Citation

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

Semplice

Travaglia

Puppo

2021

Commun. Appl. Math. Comput.

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We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. 10.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.

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“…) [1,6,23,32,44]. Results about the parameters and the accuracy of , and class reconstructions in various finite volume settings are proven in [13,15,33].…”

Section: Introductionmentioning

confidence: 94%

Section: The Novel Cwenozb Reconstruction In One Space Dimensionmentioning

confidence: 99%

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

Semplice

Travaglia

Puppo

2021

Commun. Appl. Math. Comput.

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“…Since then, these numerical schemes are being developed, modified, and extended for different fields of science and engineering, for detail see Refs. [20][21][22][23][24][25][26][27][28][29][30] Also wellbalanced finite difference and finite volume WENO schemes 6,14,[31][32][33] are developed for compressible and incompressible fluid flows.…”

Section: Introductionmentioning

confidence: 99%

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

Rehman

Ali

Zia

et al. 2021

Advances in Mechanical Engineering

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In this article, fifth order well-balanced finite volume multi-resolution weighted essentially non-oscillatory (FV MR-WENO) schemes are constructed for solving one-dimensional and two-dimensional Ripa models. The Ripa system generalizes the shallow water model by incorporating horizontal temperature gradients. The presence of temperature gradients and source terms in the Ripa models introduce difficulties in developing high order accurate numerical schemes which can preserve exactly the steady-state conditions. The proposed numerical methods are capable to exactly preserve the steady-state solutions and maintain non-oscillatory property near the shock transitions. Moreover, in the procedure of derivation of the FV MR-WENO schemes unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical schemes. Further, the results obtained from considered numerical schemes are compared with those of a high resolution central upwind scheme and available exact solutions of the Ripa model.

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“…Arbogast, Huang and Zhao in [2] introduced a new hierarchical construction that instead is capable of reducing gradually the accuracy from r l to r 1 as the discontinuity moves inward in the reconstruction stencil. WENO-AO (7,5,3) and WENO-AO (9,7,5,3) are considered in the numerical examples of the papers.…”

Section: Introductionmentioning

confidence: 99%

“…The accuracy of the CWENO3 has been studied in [23,13].The idea at the base of CWENO3, namely the use of the polynomial P 0 as in equation (2), has been exploited in different setups. Novel reconstructions of different orders of accuracy appeared under various names in the literature for the cases of one [7,3], two [8,29,17,10] and three space dimensions [37,25,38,17]. Among those, [7,8,29,38,17] consider non-uniform grids.…”

mentioning

confidence: 99%

Efficient Implementation of Adaptive Order Reconstructions

Semplice

Visconti

2020

J Sci Comput

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Including polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. In this paper we address this issue with a new Central WENOZ (CWENOZ) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the CWENOZ parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order WENO reconstructions [4,2]. The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell.Keywords. CWENOZ-AO -polynomial reconstruction -weighted essentially nonoscillatory -CWENOZadaptive order WENO -finite volume schemes -hyperbolic systemsconservation and balance laws Nq q=0 w q s(R i (t, x q )), where x q and w q are the nodes and weights of a quadrature formula on Ω i .The reconstruction operator should yield an accurate but non oscillatory pointwise approximation of the unknown, based on its cell averages. Beyond the second order of accuracy, one considers essentially non oscillatory reconstructions, which are typically realized by selecting (as in ENO [19]) or more commonly by blending in a nonlinear way polynomials with different degrees and/or stencils. In particular WENO, which was introduced in [33], considers a set of polynomials with equal degree but different stencils and aims at reproducing the accuracy of a higher degree interpolant when the data are locally smooth. The literature on this subject is vast and the reader may refer to [32] for a review.The suboptimal accuracy close to critical points shown by the original design has been later overcome by new definitions of the nonlinear weights (e.g. mapped WENO [20], WENOZ [5,9], the global average weight of [3]) or by taking the small parameter to be dependent on the local mesh size [1].Another source of difficulty in WENO reconstructions is the possible non-existence or nonpositivity of the linear weights for certain grid types and reconstruction points [31]. A quite successful proposal to address this issue was put forward by Levy, Puppo and Russo in [26] for the case of achieving a third order accurate reconstruction at cell center in one and two-dimensional uniform grids. Their CWENO3 reconstruction is a nonlinear blend of one second degree polynomial and of some first degree ones; this approach frees the linear weights from having to satisfy accuracy requirements and allows to choose them arbitrarily, independently of the reconstruction point, independently of the grid type (Ca...

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Central WENO Schemes Through a Global Average Weight

Cited by 10 publications

References 17 publications

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

Efficient Implementation of Adaptive Order Reconstructions

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