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A New Alternative WENO Scheme Based on Exponential Polynomial Interpolation with an Improved Order of Accuracy

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Abstract

In this study, we present a new alternative formulation of a conservative weighted essentially non-oscillatory (WENO) scheme that improves the performance of the known fifth-order alternative WENO (AWENO) schemes. In the formulation of the fifth-order AWENO scheme, the numerical flux can be written in two terms: a low-order flux and a high-order correction flux. The low-order numerical flux is constructed by a fifth-order WENO interpolator, and the high-order correction flux includes terms of the second and fourth derivatives, yielding the sixth-order truncation error. Noticing the difference in the convergence rates between these two approximations, this study first aims to fill the accuracy gap by enhancing the approximation order of the low-order numerical flux. To this end, the WENO interpolator for the low-order term is implemented using exponential polynomials with a shape parameter. Selecting a locally optimized shape parameter, the proposed WENO interpolator achieves an additional order of improvement, resulting in the overall sixth order of accuracy of the final reconstruction, under the same fifth-order AWENO framework. In addition, since a linear approximation to the high-order correction term may cause some oscillations in the vicinity of strong shocks, we present a new strategy for the limiting procedure to deal with the second derivative term in the high-order correction flux. Several numerical results for the well-known benchmark test problems confirm the reliability of our AWENO method.

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Acknowledgements

This research was supported by the grants NRF-2022R1F1A1066389 (H. Yang), and RS-2023-00208864, NRF-2019R1A6A1A11051177 (J. Yoon) of the National Research Foundation of Korea.

Funding

Youngsoo Ha were supported by the grant NRF-2021R1A2C1095443 through the National Research Foundation of Korea, Hyoseon Yang is supported by NRF-2022R1F1A1066389 and Jungho Yoon is supported by RS-2023-00208864 and NRF-2019R1A6A1A11051177.

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Authors and Affiliations

  1. Research Institute of Mathematics, Seoul National University, Seoul, South Korea

    Youngsoo Ha

  2. Department of Computer Engineering, Glocal Campus, Konkuk University, Chungju, South Korea

    Chang Ho Kim

  3. Department of Mathematics, Kyung Hee University, Seoul, South Korea

    Hyoseon Yang

  4. Department of Mathematics, Ewha Womans University, Seoul, South Korea

    Jungho Yoon

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  1. Youngsoo Ha
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Appendix

Appendix

  • The Lax–Friedrichs (LF) flux. The LF flux is defined by

    $$\begin{aligned} {\hat{f}}^{\textrm{LF}}_{j+1/2}(u^{-}, u^+) = \frac{1}{2}[(f(u^-) + f(u^+)) - \alpha (u^+ - u^-)], \end{aligned}$$

    where \(\alpha \) is taken as an upper bound over the whole line for \(|f'(u)|\) in the scalar case, or the absolute value of eigenvalues of the Jacobian for the system case.

  • The HLLC flux. The HLLC flux is a modified version of the HLL flux, whereby the missing contact and shear waves are restored. The HLLC flux for the Euler equations \(q=(\rho ,\rho u,E)^T\) is given by

    $$\begin{aligned} {\hat{f}}^{\textrm{HLLC}}_{j+1/2}(q^{-}, q^+) = {\left\{ \begin{array}{ll} f(q^-) &{} \text { if } 0 \le s^-, \\ f(q^-)+s^-(q^{0-} - q^-) &{} \text { if } s^- \le s^0, \\ f(q^+)+s^+(q^{0+} - q^+) &{} \text { if } s^0 \le s^+, \\ f(q^+) &{} \text { if } s^+ \le 0, \end{array}\right. } \end{aligned}$$
    (51)

    where by defining the averaging operation \(\bar{f} = \frac{1}{2}(f^+ + f^-)\) and the difference operation \(\varDelta f = f^+ - f^-\),

    $$\begin{aligned}&q^{0\pm } = \rho ^{\pm } \frac{s^{\pm } - u^{\pm }}{s^{\pm } - s^0} \begin{bmatrix} 1 \\ s^0 \\ \frac{E^{\pm }}{\rho ^{\pm }}+(s^0 - u^{\pm }) \Big (s^0 +\frac{p^{\pm }}{s^{\pm } - u^{\pm }}\Big ) \end{bmatrix}, \\&p^0 = \bar{p} - \frac{1}{2} \varDelta \bar{\rho } \bar{c}, \quad s^0 = \bar{u} - \frac{1}{2} \frac{\varDelta p}{\bar{\rho } \bar{c}}, \quad s^{\pm } = u^{\pm } \pm c^{\pm } Q^{\pm }, \\&Q^{\pm } = {\left\{ \begin{array}{ll} 1 &{} \text { if } p^0 \le p^{\pm }, \\ \Big (1 + \frac{\gamma + 1}{2 \gamma } (\frac{p^0}{p^{\pm }}-1) \Big )^{1/2} &{} \text { if }p^{\pm } \le p^0. \end{array}\right. } \end{aligned}$$

  • Ideal weights \(d_k\) based on exponential polynomials. Let \(\bar{x}= x_{j+1/2}\). The explicit forms of the Lagrange functions based on exponential polynomials in (16) are given as follows:

    $$\begin{aligned}{} & {} \ell _0(\bar{x})= \frac{3}{128} -\frac{5}{1024}(\lambda \varDelta x)^2 +\frac{65}{98304}(\lambda \varDelta x)^4 -\frac{179}{2359296}(\lambda \varDelta x)^6 + {{\mathcal {O}}}(\varDelta x^8) \nonumber \\{} & {} \ell _1(\bar{x})= -\frac{5}{32} +\frac{(\lambda \varDelta x)^2}{128} -\frac{29}{24576} (\lambda \varDelta x)^4 +\frac{53}{368640} (\lambda \varDelta x)^6 + {{\mathcal {O}}}(\varDelta x^8) \nonumber \\{} & {} \ell _2(\bar{x})= \frac{45}{64} +\frac{3}{512} \,(\lambda \varDelta x)^2 - \frac{7}{16384} (\lambda \varDelta x)^4 + \frac{47}{1966080}(\lambda \varDelta x)^6 + {{\mathcal {O}}}(\varDelta x^8) \nonumber \\{} & {} \ell _3(\bar{x})= \frac{15}{32} -\frac{1}{64}(\lambda \varDelta x)^2 +\frac{43}{24576}(\lambda \varDelta x)^4 -\frac{259}{1474560} (\lambda \varDelta x)^6 + {{\mathcal {O}}}(\varDelta x^8) \nonumber \\{} & {} \ell _4(\bar{x})= -\frac{5}{128} +\frac{7}{1024}(\lambda \varDelta x)^2 -\frac{79}{98304}(\lambda \varDelta x)^4 +\frac{989}{11796480}(\lambda \varDelta x)^6 + {{\mathcal {O}}}(\varDelta x^8). \end{aligned}$$
    (52)

    Moreover, letting \({{\bar{c}}_\lambda }= \cosh (\frac{\lambda \varDelta x}{2})\), the linear weights \({\bar{d}}_k\) based on exponential polynomials are of the form:

    $$\begin{aligned} \begin{aligned} {\bar{d}}_0&=\frac{{{\bar{c}}_\lambda }+2}{12{{\bar{c}}_\lambda }\,({{\bar{c}}_\lambda } +1)^2} = \frac{1}{16} - \frac{5}{384}(\lambda \varDelta x)^2 + {{\mathcal {O}}}(\varDelta x^4) \\ {\bar{d}}_1&=\frac{6{{\bar{c}}_\lambda }^3 +12{{\bar{c}}_\lambda }^2 +{{\bar{c}}_\lambda } -4}{6{{\bar{c}}_\lambda } ({{\bar{c}}_\lambda } +1)^2 } =\,\frac{5}{8} \, +\frac{13}{192}(\lambda \varDelta x)^2 +{{\mathcal {O}}}(\varDelta x^4) \\ {\bar{d}}_2&=\frac{3{{\bar{c}}_\lambda } +2}{4{{\bar{c}}_\lambda }\,({{\bar{c}}_\lambda } +1)^2 } =\frac{5}{16} - \frac{7}{128}(\lambda \varDelta x)^2 + {{\mathcal {O}}}(\varDelta x^4). \end{aligned} \end{aligned}$$
    (53)

  • Limiters for the correction term \({\hat{f}}^H\). To demonstrate the advantage of our limiters to the second derivative in the correction term \(\hat{f}^H\), we provide a comparison of performance between AWENO schemes with limited second derivative and central differences. In Fig. 14, we display the numerical results of the tested AWENO-E schemes for ‘Mach 3 wind tunnel problem’ in Example 4.10. We can see that the AWENO-E scheme with central differences generates more oscillations.

Fig. 14
figure 14

Density profiles of wind tunnel problem at \(t = 4\): (top) AWENO-E with limiters to the second derivative. (bottom) AWENO-E with the second-order central difference

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Ha, Y., Kim, C.H., Yang, H. et al. A New Alternative WENO Scheme Based on Exponential Polynomial Interpolation with an Improved Order of Accuracy. J Sci Comput 101, 5 (2024). https://doi.org/10.1007/s10915-024-02635-w

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