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Generalized Sensitivity Parameter Free Fifth Order WENO Finite Difference Scheme with Z-Type Weights

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Abstract

A modified fifth order Z-type (nonlinear) weights, which consist of a linear term and a nonlinear term, in the weighted essentially non-oscillatory (WENO) polynomial reconstruction procedure for the WENO-Z finite difference scheme in solving hyperbolic conservation laws is proposed. The nonlinear term is modified by a modifier function that is based on the linear combination of the local smoothness indicators. The WENO scheme with the modified Z-type weights (WENO-D) scheme and its improved version (WENO-A) scheme are proposed. They are analyzed for the maximum error and the order of accuracy for approximating the derivative of a smooth function with high order critical points, where the first few consecutive derivatives vanish. The analysis and numerical experiments show that, they achieve the optimal (fifth) order of accuracy regardless of the order of critical point with an arbitrary small sensitivity parameter, aka, satisfy the Cp-property. Furthermore, with an optimal variable sensitivity parameter, they have a quicker convergence and a significant error reduction over the WENO-Z scheme. They also achieve an improved balance between the linear term, which resolves a smooth function with the fifth order upwind central scheme, and the modified nonlinear term, which detects potential high gradients and discontinuities in a non-smooth function. The performance of the WENO schemes, in terms of resolution, essentially non-oscillatory shock capturing and efficiency, are compared by solving several one- and two-dimensional benchmark shocked flows. The results show that they perform overall as well as, if not slightly better than, the WENO-Z scheme.

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Notes

  1. A stands for Abarbanel of the late Professor Saul Abarbanel.

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Acknowledgements

The authors would like to acknowledge the funding support of this research by the National Natural Science Foundation of China (11871443), National Science and Technology Major Project (20101010), Shandong Provincial Natural Science Foundation (ZR2017MA016) and Fundamental Research Funds for the Central Universities (201562012). The author (Don) also likes to thank the Ocean University of China for providing the startup funding (201712011) that is used in supporting this work.

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Authors and Affiliations

  1. College of Oceanic and Atmospheric Sciences, Ocean University of China, Qingdao, China

    Yinghua Wang

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Bao-Shan Wang & Wai Sun Don

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  1. Yinghua Wang
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Correspondence to Wai Sun Don.

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Appendix A: Taylor Expansions of \(\phi , \tau , \beta _k\)

Appendix A: Taylor Expansions of \(\phi , \tau , \beta _k\)

The Taylor series expansions of \(\phi ^2\) at \(x_i\) is

$$\begin{aligned} \phi ^{2}= & {} \left| - \left( 2 f^{(1)}_i f^{(3)}_i \right) \Delta x^4 + \left( -\frac{1}{2} f^{(1)}_i f^{(5)}_i + \frac{13}{6} f^{(2)}_i f^{(4)}_i + \frac{7}{3} f^{(3)}_if^{(3)}_i\right) \Delta x^6 \right| \\&+\,O({\Delta x}^8) . \end{aligned}$$

The Taylor series expansions of \(\tau _5\) at \(x_i\) is

$$\begin{aligned} \tau _5= & {} \left| \left( f^{(1)}_i f^{(4)}_i-\frac{13}{3} f^{(2)}_i f^{(3)}_i \right) \Delta x^5 + \left( \frac{1}{6}f^{(1)}_if^{(6)}_i - \frac{13}{12} f^{(2)}_i f^{(5)}_i -\frac{103}{36} f^{(3)}_i f^{(4)}_i\right) \Delta x^7 \right| \\&+\,O( \Delta x^{9}). \end{aligned}$$

The Taylor series expansions of \(\beta _k\) at \(x_i\) are

$$\begin{aligned} \beta _0= & {} f^{(1)}_if^{(1)}_i \Delta x^2 + \left( \frac{13}{12} f^{(2)}_if^{(2)}_i - \frac{2}{3} f^{(1)}_i f^{(3)}_i \right) \Delta x^4 \\&- \left( \frac{13}{6} f^{(2)}_if^{(3)}_i - \frac{1}{2} f^{(1)}_i f^{(4)}_i \right) \Delta x^5 \\&-\left( \frac{7}{30} f^{(1)}_if^{(5)}_i - \frac{91}{72} f^{(2)}_i f^{(4)}_i- \frac{43}{36} f^{(3)}_if^{(3)}_i\right) \Delta x^6 \\&+\left( \frac{1}{12} f^{(1)}_if^{(6)}_i - \frac{13}{24} f^{(2)}_i f^{(5)}_i- \frac{103}{72} f^{(3)}_if^{(4)}_i\right) \Delta x^7 + O({\Delta x}^8) , \\ \beta _1= & {} f^{(1)}_if^{(1)}_i \Delta x^2 + \left( \frac{13}{12} f^{(2)}_if^{(2)}_i + \frac{1}{3} f^{(1)}_i f^{(3)}_i \right) \Delta x^4 \\&+\left( \frac{1}{60} f^{(1)}_i f^{(5)}_i+ \frac{13}{72} f^{(2)}_if^{(4)}_i + \frac{1}{36} f^{(3)}_if^{(3)}_i\right) \Delta x^6 + O( \Delta x^{8} )\\ \beta _2= & {} f^{(1)}_if^{(1)}_i \Delta x^2 + \left( \frac{13}{12} f^{(2)}_if^{(2)}_i - \frac{2}{3} f^{(1)}_i f^{(3)}_i \right) \Delta x^4 \\&+ \left( \frac{13}{6} f^{(2)}_if^{(3)}_i - \frac{1}{2} f^{(1)}_i f^{(4)}_i \right) \Delta x^5 \\&-\left( \frac{7}{30}f^{(1)}_if^{(5)}_i -\frac{91}{72} f^{(2)}_if^{(4)}_i - \frac{43}{36} f^{(3)}_if^{(3)}_i\right) \Delta x^6 \\&-\left( \frac{1}{12}f^{(1)}_if^{(6)}_i -\frac{13}{24} f^{(2)}_if^{(5)}_i - \frac{103}{72} f^{(3)}_i f^{(4)}_i \right) \Delta x^7 + O({\Delta x}^8) . \end{aligned}$$

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Wang, Y., Wang, BS. & Don, W.S. Generalized Sensitivity Parameter Free Fifth Order WENO Finite Difference Scheme with Z-Type Weights. J Sci Comput 81, 1329–1358 (2019). https://doi.org/10.1007/s10915-019-00998-z

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