New Finite Volume Weighted Essentially Nonoscillatory Schemes on Triangular Meshes

Zhu, Jun; Qiu, Jianxian

doi:10.1137/17m1112790

Cited by 56 publications

(51 citation statements)

References 31 publications

(77 reference statements)

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“…There, a polynomial of degree G ≥ 2 is combined with m = 3 (in 2D) or m = 4 (in 3D) polynomials of degree g = 1 in order to enhance the non-oscillatory properties of the schemes. At order 3 and 4, also [44] presents a similar reconstruction.…”

Section: Remarkmentioning

confidence: 71%

“…A similar definition of the nonlinear weights was employed in the CWENO context as well, obtaining the so-called CWENOZ schemes. Although sometimes under different names, [12,41,43,44] present schemes of this class. [12] compares the WENOZ and CWENOZ reconstructions, with the WENO and CWENO ones, finding that they have better spectral properties.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Cravero¹,

Semplice²,

Visconti³

2019

SIAM J. Numer. Anal.

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Central WENO reconstruction procedures have shown very good performances in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and more space dimensions, on different types of meshes. Their most recent formulations include WENOZ-type nonlinear weights, but in this context a thorough analysis of the definition of the global smoothness indicator τ is still lacking. In this work we first prove results on the asymptotic expansion of multi-dimensional Jiang-Shu smoothness indicators that are useful for the rigorous design of CWENOZ schemes also beyond those considered in this paper. Next, we introduce the optimal definition of τ for the one-dimensional CWENOZ schemes and for one example of two-dimensional CWENOZ reconstruction. Numerical experiments of one and two dimensional test problems show the good performance of the new schemes.Keywords. Central WENOZ (CWENOZ) essentially non-oscillatory reconstructions finite volume schemes smoothness indicators MSC2010. 65M08 65M20In this section we prove the theoretical results to analyse the CWENO and CWENOZ reconstruction procedures in one and more space dimensions. To this end, we restrict here to the scalar case, since usually, in the case of systems of conservation laws, the reconstruction procedures are applied component-wise, directly to the conserved variables or after the local characteristic projection.

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Section: Remarkmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Cravero¹,

Semplice²,

Visconti³

2019

SIAM J. Numer. Anal.

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show abstract

“…In order to deal with discontinuity, inspired by the existing WENO reconstruction [34], three sub-stencils S j , j = 1, 2, 3 are selected from the large stencil given in Fig. 3.…”

Section: Large Stencil and Sub-stencilsmentioning

confidence: 99%

“…where the linear weights are chosen as γ 0 = 0.97, γ 1 = γ 2 = γ 3 = 0.01 [34] without special statement.…”

Section: Define the Values Of Linear Weightsmentioning

confidence: 99%

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

Zhao

Shyy

et al. 2020

Journal of Computational Physics

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“…For a numerical flux function F compatible with f , we define F i+ 1 /2 (t) = F R i (t, x i+ 1 /2 ), R i+1 (t, x i+ 1 /2 ) and the numerical source term is computed as S i (t) = the classical hypothesis on the degree gap between high and low order polynomials when designing their reconstructions. For example, small-stencil polynomials of degree one, irrespectively of the degree of the central polynomial in [39,40,17,10]. However, in order to include very low order polynomials in the pool of candidate reconstruction polynomials, special care must be exerted.…”

Section: Introductionmentioning

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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New Finite Volume Weighted Essentially Nonoscillatory Schemes on Triangular Meshes

Cited by 56 publications

References 31 publications

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

Efficient Implementation of Adaptive Order Reconstructions

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