The 6th-order weighted ENO schemes for hyperbolic conservation laws

Hu, Fuxing

doi:10.1016/j.compfluid.2018.07.008

Cited by 13 publications

(5 citation statements)

References 23 publications

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“…Figure 27 shows the density obtained by NONS-CRCM-5 STAG-CRCM-5 and CRCM-5TVD schemes on the 960 × 240 cells. Comparing these results with those in the existing literature e.g., [7,21] it is noticed that our schemes produce the flow pattern generally accepted at present as corrected. All discontinuities are correctly positioned and well resolved.…”

Section: Example 8 Double Mach Reflection Problemsupporting

confidence: 77%

“…It is clear that our scheme can compute such sharp resolution of the complex double-blast problem, especially, since the density peaks have almost the correct value. Comparing the figures here with the results in [7,8,20], we conclude that our schemes are more accurate and more efficient. Note: For the results obtained by NONS-CRCM-5 and STAG-CRCM-5 (not presented here), they are similar to Fig.…”

Section: Example 5 Lax's Problemsupporting

confidence: 57%

“…NONS-CRCM-5 and STAG-CRCM-5 schemes are better results while the numerical solutions obtained by CRCM-5-TVD and ADER-5-TVD schemes are almost indistinguishable from the exact solution. Comparing the results here with the sixth-order WENO schemes presented in [7,8,20] (figure 5 in [8] with N = 400, figure 4 in [20] with N = 400, and figure 6 in [7] with N = 200) we observed that our schemes are comparable to them but they are much better in capturing the shocks and approximating top of the semi-ellipse. Also, we notice that the CRCM-5-TVD is superior.…”

Section: Example 2 (Example With Discontinuities)supporting

confidence: 54%

“…Figures 24, 25 and 26 show the computed density by NONS-CRCM-5 STAG-CRCM-5 and CRCM-5TVD schemes against the reference solution, which is computed by the fifth-order WENO scheme [1] with 4000 grid points. Comparing the results with ADER 5 and ADER-5-TVD schemes (figure 19-20 in [23]) and the results in [7,8,20], we observe that all the schemes presented here produce the most accurate solution and are more efficient.…”

Section: Example 7 Shock/turbulence Interaction Problemmentioning

confidence: 75%

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Central random choice methods for hyperbolic conservation laws

Zahran

Abdalla

2022

Ricerche mat

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In this article, we briefly review the random choice method (RCM) and ADER methods for solving one and two-dimensional hyperbolic conservation laws. The main advantage of RCM is that it computes discontinuities with infinite resolution. In this method, the original problem is reduced to a set of local Riemann problems (RPs). The exact solutions of these RPs are used to form the solution of the original problem. However, RCM has the following disadvantages: (1) one should solve the RP exactly, however, the exact solutions are usually complex and unavailable for many problems. (2) The accuracy of the smooth region of the flow is poor. ADER methods are explicit, one-step schemes with a very high order of accuracy in time and space. They depend on the solution of the generalized RP (GRP) exactly. In Zahran (J Math Anal Appl 346:120–140, 2008), an improved version of ADER methods (central ADER) was introduced where the RPs were solved numerically and used central fluxes, instead of upwind fluxes. The improved central ADER schemes are more accurate, faster, simple to implement, RP solver free, and need less computer memory. To fade the drawbacks of the above schemes and keep their advantages, we propose, in this paper, an improved version of the RCM. We merge the central ADER technique with the RCM. The resulting scheme is called Central RCM (CRCM). The improvements are listed as follows: we use the WENO reconstruction for the initial data instead of constant reconstruction in RCM, we solve the RPs numerically by using central finite difference schemes and use random sampling to update the solution, as the original RCM. Here we use the staggered and non-staggered RCM. To enhance the accuracy of the new methods, we use a third-order TVD flux (Zahran in Bull Belg Math Soc Simon Stevin 14:259–275, 2007), instead of a first-order flux. Compared with the original RCM and the central ADER, the new methods combine the advantages of RCM, ADER, and central finite difference methods as follows: more accurate, very simple to implement, need less computer memory, and RP solver free. Moreover, the new methods capture the discontinuities with infinite resolution and improve the accuracy of the smooth parts. The new methods have less CPU time than the central ADER methods, this is due to less flux evaluation in CRCM. An extension of the schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented. We present several numerical examples for one and two-dimensional problems. The results confirm that the presented schemes are superior to the original RCM, ADER, and central ADER schemes.

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Section: Example 8 Double Mach Reflection Problemsupporting

confidence: 77%

Section: Example 5 Lax's Problemsupporting

confidence: 57%

Section: Example 2 (Example With Discontinuities)supporting

confidence: 54%

Section: Example 7 Shock/turbulence Interaction Problemmentioning

confidence: 75%

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Central random choice methods for hyperbolic conservation laws

Zahran

Abdalla

2022

Ricerche mat

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“…The resulting schemes are high-order in smooth regions of the solution and avoid spurious oscillations near discontinuities. These schemes form an important part of the current research within numerical conservation laws, for example see [10,34,5,14,29,24,25,22]. Similarly, finite volume methods, which have the added feature of being exactly conservative, utilize flux-limiters, for example see [32,19], to switch between a high-accuracy numerical flux, and a stable numerical flux with lesser accuracy.…”

Section: Introductionmentioning

confidence: 99%

Parametric Interpolation Framework for Scalar Conservation Laws

McGregor¹,

Nave²

2019

Preprint

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In this paper we present a novel framework for obtaining high-order numerical methods for scalar conservation laws in one-space dimension for both the homogeneous and nonhomogeneous case. The numerical schemes for these two settings are somewhat different in the presence of shocks, however at their core they both rely heavily on the solution curve being represented parametrically. By utilizing high-order parametric interpolation techniques we succeed to obtain fifth order accuracy ( in space ) everywhere in the computation domain, including the shock location itself. In the presence of source terms a slight modification is required, yet the spatial order is maintained but with an additional temporal error appearing. We provide a detailed discussion of a sample scheme for non-homogeneous problems which obtains fifth order in space and fourth order in time even in the presence of shocks.

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A modified fifth‐order WENO‐Z scheme based on the weights of the reformulated adaptive order WENO scheme

Wang,

Zhao,

Yuan

2024

Numerical Methods in Fluids

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A modified fifth‐order WENO‐Z scheme is proposed by analogy with the non‐normalized weights of the reformulated fifth‐order adaptive order (AO) WENO scheme. We show that if the original fifth‐order WENO‐AO scheme is rewritten as the form of the conventional WENO combination, the resulting non‐normalized weights can be divided into three parts: a constant one term, a local stencil smoothness measure term and a global stencil smoothness measure term. In order to make use of the latter two terms for constructing a modified WENO‐Z scheme with enhanced performance, we change the form of the third term and introduce an adaptive scaling factor to adjust the contributions from the second and third terms. Numerical examples show that the modified fifth‐order WENO‐Z scheme has the advantage of high resolution in smooth regions and sharp capturing of discontinuities, and it can obtain evidently better results for shocked flows with small‐scale structures compared with the recently developed WENO‐Z+ and WENO‐Z+M schemes.

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The 6th-order weighted ENO schemes for hyperbolic conservation laws

Cited by 13 publications

References 23 publications

Central random choice methods for hyperbolic conservation laws

Central random choice methods for hyperbolic conservation laws

Parametric Interpolation Framework for Scalar Conservation Laws

A modified fifth‐order WENO‐Z scheme based on the weights of the reformulated adaptive order WENO scheme

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