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Radial Basis Function ENO and WENO Finite Difference Methods Based on the Optimization of Shape Parameters

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Abstract

We present adaptive finite difference ENO/WENO methods with infinitely smooth radial basis functions (RBFs). These methods slightly perturb the polynomial reconstruction coefficients with RBFs as the reconstruction basis and enhance accuracy in the smooth region by locally optimizing the shape parameters. Compared to the classical ENO/WENO methods, the RBF-ENO/WENO methods provide more accurate reconstructions and sharper solution profiles near the jump discontinuity. Furthermore the RBF-ENO/WENO methods are easy to implement in the existing classical ENO/WENO code. The numerical results in 1D and 2D presented in this paper show that the proposed finite difference RBF-ENO/WENO methods perform better than the classical ENO/WENO methods.

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Notes

  1. The value of \(\varepsilon ^2\) is not necessarily non-negative and \(\varepsilon \) can be either real or complex for the optimization of the error. We note that the raw value of the shape parameter, \(\varepsilon \), is not used for the reconstruction; what we use in the reconstruction is \(\varepsilon ^2\). Thus a complex value of \(\varepsilon \) is still allowed for the RBF-ENO/WENO formulation.

  2. To prevent the case that the denominator of (24) becomes zero, a small positive number \(\gamma \), is added in the denominator. In practice, we use \(\varepsilon ^2 \approx \frac{2}{\varDelta x^2}\cdot \frac{-f_{i-1}+2f_{i}-f_{i+1}}{-f_{i-1}+5f_{i}+2 f_{i+1}+\gamma }\), where \(\gamma \) is a small positive constant, e.g. \(\gamma = 10^{-13}\).

  3. Note that the denominator in (26) could be zero. However, in the implementation of the RBF-ENO/WENO methods, we apply a modified version of (27), which is explained in [8, 9]. In this alternative way of implementation, we check the magnitude of the denominator first. Only when the denominator is larger than some threshold, we compute the critical point as in (26) to check whether it is within the interval of [0, 3]. Otherwise, the stencil is considered to be smooth.

  4. The precise CFL condition is \(\varDelta t = C {\varDelta x}^{{\ell }/{3}}\), where \(\ell \) is the order of the method. Since we are using a simpler form of \(\varDelta t = C \varDelta x\), the convergence is slower than expected for the small values of N. But for the large values of N, we can use the linear relation between the temporal and spatial spacings. This is what we observe in Tables 4 and 5, as well as in Figs. 1 and 3.

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Acknowledgments

The authors thank W.-S. Don for his useful communication. We also thank the anonymous reviewers for their valuable comments.

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  1. Department of Mathematics, University at Buffalo, The State University of New York, Buffalo, NY, 14260-2900, USA

    Jingyang Guo & Jae-Hun Jung

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  1. Jingyang Guo
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Correspondence to Jae-Hun Jung.

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Guo, J., Jung, JH. Radial Basis Function ENO and WENO Finite Difference Methods Based on the Optimization of Shape Parameters. J Sci Comput 70, 551–575 (2017). https://doi.org/10.1007/s10915-016-0257-y

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