A mapping-function-free WENO-M scheme with low computational cost

Hong, Zheng; Ye, Zhengyin; Meng, Xianzong

doi:10.1016/j.jcp.2019.109145

Cited by 28 publications

(12 citation statements)

References 13 publications

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“…Recently, Don et al [9,39] demonstrated that the optimal order of accuracy for the WENO-Z scheme can be obtained regardless of any order critical points by setting the parameter ǫ, which is used to prevent the denominator becoming zero, as a function of the spacial step ∆x. However, it was indicated [21] that adjusting ǫ to recover accuracy is actually an implicit switch between the WENO scheme and the upstream central scheme, and its effect is dependent on the specific problem. It is worthy to note that, lately, Zhu et al [46] has proposed a new type of high-order MWENO schemes with unequal-sized stencils to achieve increasingly high order of accuracy for the seventh-order and ninthorder versions.…”

Section: Introductionmentioning

confidence: 99%

“…For the WENO-PM6 scheme, another concerned WENO scheme in this paper, it was noted that [38] the piecewise nature requires logic operations to be implemented during the mapping process, which makes it harder to vectorize the operations and increases the computational cost. Hong et al indicated that [21], as the form of the mapping function designed in the WENO-PM6 scheme becomes more complicated than the WENO-M scheme, its extra computational cost will have a further increase. Actually, our extensive numerical tests show that the extra computational cost of the WENO-PM6 scheme will increase by more than 54% compared to the WENO-JS scheme, and we will give the results in Subsection 4.4 in detail.…”

Section: Introductionmentioning

confidence: 99%

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An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

Li¹,

Zhong²

2022

NMTMA

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We present a novel mapping approach for WENO schemes through the use of an approximate constant mapping function which is constructed by employing an approximation of the classic signum function. The new approximate constant mapping function is designed to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was proposed to overcome the potential loss of accuracy of the WENO-M scheme which was developed to recover the optimal convergence order of the WENO-JS scheme at critical points. Our new mapped WENO scheme, denoted as WENO-ACM, maintains almost all advantages of the WENO-PM6 scheme, including low dissipation and high resolution, while decreases the number of mathematical operations remarkably in every mapping process leading to a significant improvement of efficiency. The convergence rates of the WENO-ACM scheme have been shown through one-dimensional linear advection equation with various initial conditions. Numerical results of one-dimensional Euler equations for the Riemann problems, the Mach 3 shock-density wave interaction and the Woodward-Colella interacting blastwaves are improved in comparison with the results obtained by the WENO-JS, WENO-M and WENO-PM6 schemes. Numerical experiments with two-dimensional problems as the 2D Riemann problem, the shock-vortex interaction, the 2D explosion problem, the double Mach reflection and the forward-facing step problem modeled via the two dimensional Euler equations have been conducted to demonstrate the high resolution and the effectiveness of the WENO-ACM scheme. The WENO-ACM scheme provides significantly better resolution than the WENO-M scheme and slightly better resolution than the WENO-PM6 scheme, and compared to the WENO-M and WENO-PM6 schemes, the extra computational cost is reduced by more than 83% and 93%, respectively.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

Li¹,

Zhong²

2022

NMTMA

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“…In order to reduce the cost of mapping process, Hong et al [30] proposed a pre-discrete mapping method.…”

Section: Pre-discrete Mapping Methodsmentioning

confidence: 99%

An Improved WENO-Z Scheme with Symmetry-Preserving Mapping

Hong

Ye²,

2020

Preprint

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Abstract Since the classical weighted essentially non-oscillatory (WENO) scheme is proposed, various improved versions have been developed, and typical one is the WENO-Z scheme. Although better resolution is achieved, it is shown in this article that, the result of WENO-Z scheme suffers evident distortion in the long-time simulation of the linear advection equation. In order to fix the problem of WENO-Z, a symmetry-preserving mapping method is proposed in this article. In the original mapping method, the weight of each sub-stencil is used to map, which is demonstrated to cause asymmetric improvement about a discontinuity. This asymmetric improvement will lead to a distorted solution, more severe with longer output time. In the symmetry-preserving mapping method, a new variable related to the smoothness indicator is selected to map, which has the same ideal value for each sub-stencil. Using the new mapping method can not only fix the distortion problem of WENO-Z, but also improve the numerical resolution. Several benchmark problems are conducted to show the improved performance of the resultant scheme.

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“…In order to reduce the cost of mapping process, Hong et al [30] proposed a prediscrete mapping method. The method consists of three steps, which are described as follows.…”

Section: Pre-discrete Mapping Methodsmentioning

confidence: 99%

An improved WENO-Z scheme with symmetry-preserving mapping

Hong

2020

Adv. Aerodyn.

Self Cite

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Since the classical weighted essentially non-oscillatory (WENO) scheme is proposed, various improved versions have been developed, and a typical one is the WENO-Z scheme. Although better resolution is achieved, it is shown in this article that, the result of WENO-Z scheme suffers evident distortion in the long-time simulation of the linear advection equation. In order to fix the problem of WENO-Z, a symmetrypreserving mapping method is proposed in this article. In the original mapping method, the weight of each sub-stencil is used to map, which is demonstrated to cause asymmetric improvement about a discontinuity. This asymmetric improvement will lead to a distorted solution, more severe with longer output time. In the symmetry-preserving mapping method, a new variable related to the smoothness indicator is selected to map, which has the same ideal value for each sub-stencil. Using the new mapping method can not only fix the distortion problem of WENO-Z, but also improve the numerical resolution. Several benchmark problems are conducted to show the improved performance of the resultant scheme.

show abstract

A mapping-function-free WENO-M scheme with low computational cost

Cited by 28 publications

References 13 publications

An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

An Improved WENO-Z Scheme with Symmetry-Preserving Mapping

An improved WENO-Z scheme with symmetry-preserving mapping

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