A New Mapped Weighted Essentially Non-oscillatory Scheme

Feng, Hui; Hu, Fuxing; Wang, Rong

doi:10.1007/s10915-011-9518-y

Cited by 85 publications

(121 citation statements)

References 19 publications

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“…Namely, a high value for may yield oscillatory reconstructions, while low values yields an estimated order of convergence that matches the theoretical one only asymptotically and often only for very fine grid spacings. Among the different solutions proposed in the literature, the mappings of [16,13] do not apply in a straightforward way to our compact WENO reconstruction technique, while taking an h-dependent as in [2] yields an improvement of the reconstruction, but we found experimentally that the scaling ∝ h works best in our situation (see the numerical tests). Our choice is further supported by the work of Kolb [21] that analyzed the optimal convergence rate of CWENO schemes depending on the choice of (h) on uniform schemes.…”

Section: One Space Dimensionmentioning

confidence: 91%

Adaptive Mesh Refinement for Hyperbolic Systems Based on Third-Order Compact WENO Reconstruction

2015

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In this paper we generalize to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput., 2001), thus obtaining a truly two-dimensional non-oscillatory third order reconstruction with a very compact stencil and that does not involve mesh-dependent coefficients. This latter characteristic is quite valuable for its use in h-adaptive numerical schemes, since in such schemes the coefficients that depend on the disposition and sizes of the neighboring cells (and that are present in many existing WENO-like reconstructions) would need to be recomputed after every mesh adaption.In the second part of the paper we propose a third order h-adaptive scheme with the above-mentioned reconstruction, an explicit third order TVD RungeKutta scheme and the entropy production error indicator proposed by Puppo and Semplice (Commun. Comput. Phys., 2011). After devising some heuristics on the choice of the parameters controlling the mesh adaption, we demonstrate with many numerical tests that the scheme can compute numerical solution whose error decays as N −3 , where N is the average number of cells used during the computation, even in the presence of shock waves, by making a very effective use of h-adaptivity and the proposed third order reconstruction.

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Section: One Space Dimensionmentioning

confidence: 91%

Adaptive Mesh Refinement for Hyperbolic Systems Based on Third-Order Compact WENO Reconstruction

2015

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“…Based on these errors, it is confirmed that there is no numerical instability with the WENO-MN3 scheme. Note that we have made performance analysis of WENO-MN3 along with the WENO-JS scheme but not with the mapped WENO, as these calculations can be used to verify the results presented in the works of Feng et al 16…”

Section: Nonlinear Discontinuity Problemmentioning

confidence: 81%

“…Example Consider the transport equation with wave speed 1 along with the initial conditions

u_{0} false(x false) = \sin^{9} false(π x false),

and

u_{0} false(x false) = ({, \begin{matrix} array \\ array 1, & array - 1 < x < 0, \\ array 0, & array 0 \leq x \leq 1 . \end{matrix})

We have calculated the L 1 ‐ and L ∞ ‐errors for WENO‐JS scheme with ε =10 −6 along with the WENO‐MN3 scheme with ε =Δ x 2 presented in Tables and , respectively. Based on these errors, it is confirmed that there is no numerical instability with the WENO‐MN3 scheme.…”

Section: Numerical Resultsmentioning

confidence: 99%

Improved third‐order weighted essentially nonoscillatory scheme

Gande

Rathod

Rathan

2018

Numerical Methods in Fluids

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Summary A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.

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“…If q = 2, this is sufficient to satisfy the consistency condition. If q > 2, for instance for WENO-Z, the coefficients a kl must also provide linear combinations of smoothness indicators that cancel out the lower order terms in the Taylor expansions (10). To achieve this, the general form (26) is adjusted.…”

Section: General Frameworkmentioning

confidence: 99%

“…Recently, several variants of the WENO scheme have appeared that improve the order of accuracy near points where the first derivative vanishes. Examples include the WENO-M [9,10], WENO-Z [2,11,12] and WENO-NS [13] schemes. For a comparison of the performance of these schemes, see Zhao et al [14].…”

Section: Introductionmentioning

confidence: 99%

Embedded WENO: A design strategy to improve existing WENO schemes

Lith

Boonkkamp

IJzerman

2017

Journal of Computational Physics

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Embedded WENO methods utilize all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes under-use this possibility close to large gradients or discontinuities. We develop a general approach for constructing embedded versions of existing WENO schemes. Embedded methods based on the WENO schemes of Jiang and Shu [1] and on the WENO-Z scheme of Borges et al.[2] are explicitly constructed. Several possible choices are presented that result in either better spectral properties or a higher order of convergence for sufficiently smooth solutions. However, these improvements carry over to discontinuous solutions. The embedded methods are demonstrated to be indeed improvements over their standard counterparts by several numerical examples. All the embedded methods presented have no added computational effort compared to their standard counterparts.

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A New Mapped Weighted Essentially Non-oscillatory Scheme

Cited by 85 publications

References 19 publications

Adaptive Mesh Refinement for Hyperbolic Systems Based on Third-Order Compact WENO Reconstruction

Adaptive Mesh Refinement for Hyperbolic Systems Based on Third-Order Compact WENO Reconstruction

Improved third‐order weighted essentially nonoscillatory scheme

Embedded WENO: A design strategy to improve existing WENO schemes

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