Third‐ and fourth‐order well‐balanced schemes for the shallow water equations based on the CWENO reconstruction

Castro, Manuel J.; Semplice, Matteo

doi:10.1002/fld.4700

Cited by 29 publications

(29 citation statements)

References 39 publications

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“…Example 4 (2D of order 4 on cartesian meshes). CWENO reconstructions of order 4 on uniform cartesian meshes in two space dimensions were introduced in [11]. There, a polynomial of degree G = 3 defined by a diamond-shaped central stencil is combined with m == 4 polynomials of degree g = 1 or g = 2.…”

Section: Remarkmentioning

confidence: 99%

“…where the parameters t, s, u can take any real values. In these last cases we have obtained aτ of the same order as the τ in [12] and, the global smoothness indicators of [12] correspond to the choice t = 1 inτ 7 , u = 0, t = 1 inτ 9 and u = 0, t = 1, s = t inτ 11 .…”

Section: Cwenoz In One Spatial Dimensionmentioning

confidence: 99%

“…4 (x) the analogous polynomials in the south-east, south-west or north-west sub-stencils respectively. The expression for all the polynomials involved is given in [11] and is reported here for convenience.…”

Section: Cwenoz3 In Two Spatial Dimensionmentioning

confidence: 99%

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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: 99%

Section: Cwenoz In One Spatial Dimensionmentioning

confidence: 99%

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Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Cravero¹,

Semplice²,

Visconti³

2019

SIAM J. Numer. Anal.

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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%

“…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.…”

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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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Finite difference modified WENO schemes for hyperbolic conservation laws with non‐convex flux

Dond

Kumar

2021

Numerical Methods in Fluids

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The Weighted Essentially Non-Oscillatory (WENO) reconstruction provides higher-order accurate solutions to hyperbolic conservation laws for convex flux.But it fails to capture composite structure in the case of non-convex flux and converges to the wrong solution (Qiu & Shu SIAM J Sci Comput. 2008;31:584-607). In this article, we have developed a Modified WENO (MWENO) scheme in the finite difference framework, which can resolve the composite structure and ensures the entropic convergence. The MWENO reconstruction procedure involves the identification of the troubled-cells, followed by the use of first-order monotone modification in the troubled-cells and employ the fifth-order WENO reconstruction in the non-troubled cells. A new troubled-cell indicator is developed using the information of the smoothness indicator of the WENO reconstruction. Numerical experiments are performed for 1D and 2D test cases, which ensure the entropic convergence of the proposed schemes.

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Third‐ and fourth‐order well‐balanced schemes for the shallow water equations based on the CWENO reconstruction

Cited by 29 publications

References 39 publications

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Efficient Implementation of Adaptive Order Reconstructions

Finite difference modified WENO schemes for hyperbolic conservation laws with non‐convex flux

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