On the Full and Global Accuracy of a Compact Third Order WENO Scheme

Kolb, Oliver

doi:10.1137/130947568

Cited by 56 publications

(63 citation statements)

References 24 publications

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“…Within the following numerical examples, we consider two different discretization schemes. The first is a third-order CWENO scheme (CWENO3) with suitable boundary treatment [20,23], which relies on a local Lax-Friedrichs flux function for the inner discretization points of each pipe and handles coupling points by explicitly solving equation (27). The second scheme is an implicit box scheme (IBOX) [21], suitable for sub-sonic flows.…”

Section: Numerical Resultsmentioning

confidence: 99%

Modeling and simulation of gas networks coupled to power grids

2019

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In this paper, a mathematical framework for the coupling of gas networks to electric grids is presented to describe in particular the transition from gas to power. The dynamics of the gas flow are given by the isentropic Euler equations, while the power flow equations are used to model the power grid. We derive pressure laws for the gas flow that allow for the wellposedness of the coupling and a rigorous treatment of solutions. For simulation purposes, we apply appropriate numerical methods and show in a experimental study how gas-to-power might influence the dynamics of the gas and power network, respectively.

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Section: Numerical Resultsmentioning

confidence: 99%

Modeling and simulation of gas networks coupled to power grids

2019

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“…The presence of a positive ϵ at the denominator in the previous formula is essential in order to avoid a division by zero in the case of reconstruction of very flat data. However, in the one‐dimensional case it was proven (for both WENO and CWENO) that the value of ϵ can affect the convergence rate close to local extrema. Intuitively, when a local extrema is present in the stencil, some of the indicators will be of size

scriptO false(h^{4} false)

and some others will be ≈ h 2 : unless ϵ is at least as big as h 2 , formula will bias the nonlinear weights toward the lower‐order polynomials with smaller indicators, effectively leading to an order reduction.…”

Section: Central Weighted Essentially Nonoscillatory Reconstruction (mentioning

confidence: 99%

“…After the seminal paper, the one‐dimensional CWENO technique was extended to fifth order, the properties of the third‐order versions were studied in detail on uniform meshes and nonuniform ones, and finally, arbitrary high‐order variants were introduced …”

Section: Introductionmentioning

confidence: 99%

“…After the seminal paper, 11 the one-dimensional CWENO technique was extended to fifth order, 13,14 the properties of the third-order versions were studied in detail on uniform meshes 15 and nonuniform ones, 16 and finally, arbitrary high-order variants were introduced. 17,18 The CWENO procedure in more than one space dimension has been first employed in the form of a combination of a central parabola in two variables and four linear polynomials.…”

Section: Introductionmentioning

confidence: 99%

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

Castro

Semplice

2018

Numerical Methods in Fluids

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Summary High‐order finite volume schemes for conservation laws are very useful in applications, due to their ability to compute accurate solutions on quite coarse meshes and with very few restrictions on the kind of cells employed in the discretization. For balance laws, the ability to approximate up to machine precision relevant steady states allows the scheme to compute accurately, also on coarse meshes, small perturbations of such states, which are very relevant for many applications. In this paper, we propose third‐ and fourth‐order accurate finite volume schemes for the shallow water equations. The schemes have the well‐balanced property thanks to a path‐conservative approach applied to an appropriate nonconservative reformulation of the equations. High‐order accuracy is achieved by designing truly two‐dimensional (2D) reconstruction procedures of the central WENO (CWENO) type. The novel schemes are tested for accuracy and well‐balancing and shown to maintain positivity of the water height on wet/dry transitions. Finally, they are applied to simulate the Tohoku 2011 tsunami event.

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“…Consequently no issues regarding their existence and positivity is present. 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].…”

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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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On the Full and Global Accuracy of a Compact Third Order WENO Scheme

Cited by 56 publications

References 24 publications

Modeling and simulation of gas networks coupled to power grids

Modeling and simulation of gas networks coupled to power grids

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

Efficient Implementation of Adaptive Order Reconstructions

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