A new type of multi-resolution WENO schemes with increasingly higher order of accuracy

Zhu, Jun; Shu, Chi-Wang

doi:10.1016/j.jcp.2018.09.003

Cited by 138 publications

(86 citation statements)

References 52 publications

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“…Even though we have considered the finite volume formulation, we believe that the same results extend straightforwardly to the finite difference case. Furthermore, also the recently proposed Multiresolution WENO schemes [41,42], which are based on a hierarchical computation for the nonlinear weights, might be amenable to be sped up following the ideas of this paper.…”

Section: Resultsmentioning

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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Section: Resultsmentioning

confidence: 99%

Efficient Implementation of Adaptive Order Reconstructions

Semplice

Visconti

2020

J Sci Comput

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“…The two-dimensional hyperbolic conservation laws (2) are used as an example to explain the new fifth-order, seventh-order, and ninth-order finite-difference multi-resolution WENO schemes [54]. Thus we take the associated semidiscretization of (2) and reformulate it as where L(u) is the high-order spatial discretization of −f (u) x − g(u) y .…”

Section: Finite-difference Multi-resolution Weno Schemesmentioning

confidence: 99%

“…Very recently, a new type of high-order multi-resolution WENO schemes has been designed to solve time dependent hyperbolic conservation laws on structured meshes [54]. We design this new type of multi-resolution WENO schemes only borrowing the original idea of the multi-resolution methods [19][20][21][22][23] without introducing the multi-resolution representation of the solution and data compression (i.e., encoding/decoding algorithms).…”

Section: Introductionmentioning

confidence: 99%

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Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

Zhu

Chi

Wang³

2019

Commun. Appl. Math. Comput.

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A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659-683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202-228, 1996) might suffer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain fifth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state problems. In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

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“…These strategies can be applied to other types of weighted schemes as well [12,14]. In more recent years, Zhu et al [16][17][18][19][20][21][22] proposed new types of WENO schemes with the application of a series of unequal-sized spatial stencils. Such new WENO schemes have very nice convergence property to steady state solutions.…”

Section: Introductionmentioning

confidence: 99%

A brief review on the convergence to steady state solutions of Euler equations with high-order WENO schemes

2019

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Weighted essentially non-oscillatory (WENO) schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions. However, like many other high-order shock capturing schemes, WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave. This is a long-standing difficulty for high-order shock capturing schemes. In recent years, this non-convergence problem has been studied extensively for WENO schemes. Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations, which are at the small local truncation error level but prevent the residue to settle down to machine zero. Several strategies have been proposed to reduce these slight post shock oscillations, including the design of new smoothness indicators for the fifth-order WENO scheme, the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy, and the design of a new type of WENO schemes. With these strategies, the convergence to steady states is improved significantly. Moreover, the strategies are applicable to other types of weighted schemes. In this paper, we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.

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A new type of multi-resolution WENO schemes with increasingly higher order of accuracy

Cited by 138 publications

References 52 publications

Efficient Implementation of Adaptive Order Reconstructions

Efficient Implementation of Adaptive Order Reconstructions

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A brief review on the convergence to steady state solutions of Euler equations with high-order WENO schemes

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