A Sixth-Order Weighted Essentially Non-oscillatory Schemes Based on Exponential Polynomials for Hamilton–Jacobi Equations

Ha, Youngsoo; Kim, Chang Ho; Yang, Hyoseon; Yoon, Jungho

doi:10.1007/s10915-017-0603-8

Cited by 13 publications

(12 citation statements)

References 50 publications

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“…with the coefficients a k obtained by evaluating the integral at the stencil nodes [13,19]. Equivalently, a convenient way to construct the numerical flux f is via Lagrange's interpolation formula to the primitive function H of h on the cell-boundaries (say…”

Section: Interpolation Based On Exponential Polynomial Basis Functionsmentioning

confidence: 99%

“…Some other fifth-order WENO schemes were further proposed by modifying nonlinear weights [9,12,25,49]. Sixth or higher order WENO techniques have been developed in the literature [3,10,13,20,21]. The central WENO [5,24,27], hybrid compact WENO schemes [32,39], and other versions of the WENO methods [2,6,27,28,31,46,50] have been constructed to improve the performance of the WENO techniques.…”

Section: Introductionmentioning

confidence: 99%

“…For a given exponential polynomial space, the choice of the tension (or shape) parameter λ has a significant impact on the accuracy of interpolation. A well-selected parameter can yield better results compared to the polynomialbased method for various types of PDEs [11,13,45,47,48]. However, selecting an optimal parameter is an important and difficult problem, indeed a topic of active research.…”

Section: Introductionmentioning

confidence: 99%

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Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space

Kim

Yang

et al. 2020

Preprint

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The aim of this study is to develop a novel WENO scheme that improves the performance of the well-known fifth-order WENO methods. The approximation space consists of exponential polynomials with a tension parameter that may be optimized to fit the the specific feature of the data, yielding better results compared to the polynomial approximation space. However, finding an optimal tension parameter is a very important and difficult problem, indeed a topic of active research. In this regard, this study introduces a practical approach to determine an optimal tension parameter by taking into account the relationship between the tension parameter and the accuracy of the exponential polynomial interpolation under the setting of the fifth-order WENO scheme. As a result, the proposed WENO scheme attains an improved order of accuracy (that is, sixth-order) better than other fifth-order WENO methods without loss of accuracy at critical points. A detailed analysis is provided to verify the improved convergence rate. Further, we present modified nonlinear weights based on L 1 -norm approach along with a new global smoothness indicator. The proposed nonlinear weights reduce numerical dissipation significantly, while attaining better resolution in smooth regions. Some experimental results for various benchmark test problems are presented to demonstrate the ability of the new scheme.

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Section: Interpolation Based On Exponential Polynomial Basis Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space

Kim

Yang

et al. 2020

Preprint

Self Cite

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“…In order to address this problem, schemes were developed based on both trigonometric [47,48] and exponential [16,18] functions in the interpolation basis for ENO and WENO. This paper proposes to use non-polynomial function approximations by formulating a WENO scheme in terms of a RBF and achieving superconvergence by tuning the available shape or tension parameter.…”

Section: Rbfs φ(X)mentioning

confidence: 99%

“…In [16], the authors introduced a WENO scheme based on the space of exponential polynomials. Later, in the work [18], they improved the order of accuracy of their schemes by exploiting the control parameter λ ∈ R or iR for exponential basis functions of the form e λx .…”

Section: Optimal Shape Parameters For Rbfsmentioning

confidence: 99%

Superconvergent Non-Polynomial Approximations

Christlieb¹,

Sands²,

Yang³

2020

Preprint

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In this paper, we introduce a superconvergent approximation method using a non-polynomial that consists of radial basis functions (RBFs) to solve conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at the cell interfaces. Superconvergence of the approximations is addressed through analyses, which result in the attainment of expressions for optimal RBF shape parameters. This study seeks to address the practical elements of the approach, including the evaluations of shape parameters and a hybrid implementation. To highlight the effectiveness of the non-polynomial basis, the proposed methods are applied to one-dimensional hyperbolic and weakly hyperbolic systems of conservation laws. In the latter case, notable improvements are observed in predicting the location and height of the finite time blowup. The convergence results demonstrate that the proposed schemes attain improvements in accuracy, as indicated by the analyses.

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Exponential approximation space reconstruction weighted essentially nonoscillatory scheme for dispersive partial differential equations

Salian,

Samala

2023

Math Methods in App Sciences

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In this work, we construct a fifth‐order weighted essentially non‐oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third‐order derivative formulation is developed directly using WENO spatial reconstruction procedure, and third‐order TVD Runge–Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the characteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the construction of conservative flux approximation, smoothness indicators, nonlinear weights, and verified that the proposed scheme provides the required fifth convergence order. One‐ and two‐dimensional numerical examples are presented to support the theoretical claims.

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A Sixth-Order Weighted Essentially Non-oscillatory Schemes Based on Exponential Polynomials for Hamilton–Jacobi Equations

Cited by 13 publications

References 50 publications

Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space

Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space

Superconvergent Non-Polynomial Approximations

Exponential approximation space reconstruction weighted essentially nonoscillatory scheme for dispersive partial differential equations

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