Improving Accuracy of the Fifth-Order WENO Scheme by Using the Exponential Approximation Space

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

doi:10.1137/20m1317396

Cited by 8 publications

(2 citation statements)

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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 . We adopt a similar strategy in this work with the difference being the choice of the basis.…”

Section: Optimal Shape Parameters For Rbfsmentioning

confidence: 99%

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Superconvergent Non-Polynomial Approximations

Christlieb¹,

Sands²,

Yang³

2020

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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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Section: Rbfs φ(X)mentioning

confidence: 99%

Section: Optimal Shape Parameters For Rbfsmentioning

confidence: 99%