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An Essential Seventh-Order Weighted Compact Adaptive Scheme for Hyperbolic Conservation Laws

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Abstract

A new seventh-order weighted compact adaptive scheme is proposed in this paper. The proposed reconstruction is inspired by the upwind compact scheme and the adaptive order weighted essentially non-oscillatory (WENO) scheme. The proposed seventh-order scheme is a convex combination of a linear seventh-order compact scheme and three linear third-order compact schemes. The reconstruction of the proposed scheme is based on the same stencil as the reconstruction of the original fifth-order weighted compact scheme. Various classical tests are presented to show the performance of the proposed numerical scheme. The numerical results of the proposed scheme are compared with those obtained with the original fifth-order compact WENO scheme, seventh-order compact WENO scheme and classical WENO schemes. The numerical results show that the proposed scheme preserves the designed seventh-order of accuracy in smooth regions and produces higher resolution solutions near shocks or discontinuities without numerical oscillations with less computational cost than the original weighted compact schemes.

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References

  1. Adams, N.A., Shariff, K.: A high-resolution hybrid compact-ENO scheme for shock–turbulence interaction problems. J. Comput. Phys. 127(1), 27–51 (1996)

    Article  MathSciNet  Google Scholar 

  2. Arbogast, T., Huang, C.-S., Zhao, X.: Accuracy of WENO and adaptive order WENO reconstructions for solving conservation laws. SIAM J. Numer. Anal. 56(3), 1818–1847 (2018)

    Article  MathSciNet  Google Scholar 

  3. Balsara, D.S., Garain, S., Florinski, V., Boscheri, W.: An efficient class of WENO schemes with adaptive order for unstructured meshes. J. Comput. Phys. 404, 109062 (2020)

    Article  MathSciNet  Google Scholar 

  4. Balsara, D.S., Garain, S., Shu, C.-W.: An efficient class of WENO schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016)

    Article  MathSciNet  Google Scholar 

  5. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000)

    Article  MathSciNet  Google Scholar 

  6. Batten, P., Clarke, N., Lambert, C., Causon, D.M.: On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18(6), 1553–1570 (1997)

    Article  MathSciNet  Google Scholar 

  7. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)

    Article  MathSciNet  Google Scholar 

  8. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  Google Scholar 

  9. Chen, Y., Jiang, S., Liu, N.: HFVS: an arbitrary high order approach based on flux vector splitting. J. Comput. Phys. 322, 708–722 (2016)

    Article  MathSciNet  Google Scholar 

  10. Deng, X., Zhang, H.: Developing high-order weighted compact nonlinear schemes. J. Comput. Phys. 165(1), 22–44 (2000)

    Article  MathSciNet  Google Scholar 

  11. Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate Fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)

    Article  MathSciNet  Google Scholar 

  12. Gao, Z., Wen, X., Don, W.S.: Enhanced robustness of the hybrid compact-WENO finite difference scheme for hyperbolic conservation laws with multi-resolution analysis and Tukey’s boxplot method. J. Sci. Comput. 73(2), 736–752 (2017)

  13. Gerolymos, G.A., Sénéchal, D., Vallet, I.: Very-high-order WENO schemes. J. Comput. Phys. 228(23), 8481–8524 (2009)

    Article  MathSciNet  Google Scholar 

  14. Ghosh, D., Baeder, J.D.: Compact reconstruction schemes with weighted ENO limiting for hyperbolic conservation laws. SIAM J. Sci. Comput. 34(3), 1678–1706 (2012)

    Article  MathSciNet  Google Scholar 

  15. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)

    Article  MathSciNet  Google Scholar 

  16. Guo, Y., Shi, Y.F.: Seventh order compact-WENO scheme for hyperbolic conservation laws. Comput. Fluids 176, 193–209 (2018)

    Article  MathSciNet  Google Scholar 

  17. Guo, Y., Xiong, T., Shi, Y.: A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations. J. Comput. Phys. 274, 505–523 (2014)

    Article  MathSciNet  Google Scholar 

  18. Guo, Y., Xiong, T., Shi, Y.: A maximum-principle-satisfying high-order finite volume compact WENO scheme for scalar conservation laws with applications in incompressible flows. J. Sci. Comput. 65(1), 83–109 (2015)

    Article  MathSciNet  Google Scholar 

  19. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983)

    Article  MathSciNet  Google Scholar 

  20. Huang, C., Chen, L.L.: A simple smoothness indicator for the WENO scheme with adaptive order. J. Comput. Phys. 352, 498–515 (2018)

    Article  MathSciNet  Google Scholar 

  21. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  Google Scholar 

  22. Jiang, L., Shan, H., Liu, C.: Weighted compact scheme for shock capturing. Int. J. Comput. Fluid Dyn. 15(2), 147–155 (2001)

    Article  MathSciNet  Google Scholar 

  23. Kumar, R., Chandrashekar, P.: Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws. J. Comput. Phys. 375, 1059–1090 (2018)

    Article  MathSciNet  Google Scholar 

  24. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MathSciNet  Google Scholar 

  25. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22(2), 656–672 (2000)

    Article  MathSciNet  Google Scholar 

  26. Li, P., Don, W.-S., Wang, C., Gao, Z.: High order positivity- and bound-preserving hybrid compact-WENO finite difference scheme for the compressible Euler equations. J. Sci. Comput. 74(2), 640–666 (2018)

    Article  MathSciNet  Google Scholar 

  27. Liska, R., Wendroff, B.: Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25(3), 995–1017 (2003)

    Article  MathSciNet  Google Scholar 

  28. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  Google Scholar 

  29. Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock–turbulence interaction. J. Comput. Phys. 178(1), 81–117 (2002)

    Article  MathSciNet  Google Scholar 

  30. Ren, Y.-X., Liu, M., Zhang, H.: A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 192(2), 365–386 (2003)

    Article  MathSciNet  Google Scholar 

  31. Schulz-Rinne, C.W., Collins, J.P., Glaz, H.M.: Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput. 14(6), 1394–1414 (1993)

    Article  MathSciNet  Google Scholar 

  32. Semplice, M., Visconti, G.: Efficient implementation of adaptive order reconstructions. J. Sci. Comput. 83(1), 6 (2020)

    Article  MathSciNet  Google Scholar 

  33. Shen, Y., Yang, G.: Hybrid finite compact-WENO schemes for shock calculation. Int. J. Numer. Methods Fluids 53(4), 531–560 (2007)

    Article  MathSciNet  Google Scholar 

  34. Titarev, V.A., Toro, E.F.: Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238–260 (2004)

    Article  MathSciNet  Google Scholar 

  35. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics (2009)

  36. Wang, B.S., Don, W.S., Gao, Z., Wang, Y.H., Wen, X.: Hybrid compact-WENO finite difference scheme with radial basis function based shock detection method for hyperbolic conservation laws. SIAM J. Sci. Comput. 40(6), A3699–A3714 (2018)

    Article  MathSciNet  Google Scholar 

  37. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)

    Article  MathSciNet  Google Scholar 

  38. Zhang, S., Jiang, S., Shu, C.-W.: Development of nonlinear weighted compact schemes with increasingly higher order accuracy. J. Comput. Phys. 227(15), 7294–7321 (2008)

    Article  MathSciNet  Google Scholar 

  39. Zhao, G.-Y., Sun, M.-B., Pirozzoli, S.: On shock sensors for hybrid compact/WENO schemes. Comput. Fluids 199, 104439 (2020)

    Article  MathSciNet  Google Scholar 

  40. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    Article  MathSciNet  Google Scholar 

  41. Zhu, J., Qiu, J.: A new type of finite volume WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 73(2), 1338–1359 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (2020QN01).

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Authors and Affiliations

  1. School of Electric and Power Engineering, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    YuFeng Shi

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Yan Guo

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  1. YuFeng Shi
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Correspondence to Yan Guo.

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Appendix The HLLC flux

Appendix The HLLC flux

The HLLC flux for the Euler equations is given by [6, 35]

$$\begin{aligned} {\hat{\mathbf{H }}}^{HLLC}(U_l, U_r)=\left\{ \begin{array}{ll} F_l, &{}\quad \text {if }S_L>0,\\ F^{*}_l=F_l+S_l(U^{*}_l-U_l), &{}\quad \text {if }S_L\le 0<S_M,\\ F^{*}_r=F_r+S_r(U^{*}_r-U_r), &{}\quad \text {if }S_M\le 0\le S_R,\\ F_r, &{}\quad \text {if }S_R<0. \end{array}\right. \end{aligned}$$

where \(F_l=F(U_l)\) and \(F_r=F(U_r)\), \(U_l\) and \(U_r\) denote the approximate Riemann solution, \(U^{*}_l, U^{*}_r\) denote two averaged states between the two acoustic waves \(S_L, S_R\). The acoustic wavespeeds are computed from

$$\begin{aligned} S_L=\min {[u_l-a_l,{\tilde{u}}^{*}-{\tilde{a}}^{*}]}, \quad S_R=\min {[u_r+a_r,{\tilde{u}}^{*}+{\tilde{a}}^{*}]}, \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{l} {\tilde{u}}^{*}=\frac{u_l+u_rR_\rho }{1+R_\rho },\\ {\tilde{a}}^{*}=\sqrt{(\gamma -1)[{\tilde{H}}^{*}-\frac{1}{2}{\tilde{u}}^{*2}]},\\ {\tilde{H}}^{*}=\frac{(H_l+H_rR_\rho )}{1+R_\rho },\\ R_\rho =\sqrt{\frac{\rho _r}{\rho _l}}, \end{array}\right. \end{aligned}$$

where \(H=\frac{e+p}{\rho }\) is the enthalpy. The left star states \(U^{*}_l\) can be obtained as follows

$$\begin{aligned} \left\{ \begin{array}{l} \rho ^{*}_l=\rho _l\frac{S_L-u_l}{S_L-S_M},\\ p^{*}=\rho _l(u_l-S_L)(u_l-S_M)+p_l,\\ \rho ^{*}_lu^{*}_l=\frac{(S_L-u_l)\rho _lu_l+(p^{*}-p_l)}{S_L-S_M},\\ E^{*}_l=\frac{(S_L-u_l)E_l+p^{*}S_M}{S_L-S_M}, \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} S_M=\frac{\rho _ru_r(S_R-u_r)-\rho _lu_l(S_L-u_l)+p_l-p_r}{\rho _l(S_R-u_r)-\rho _l(S_L-u_l)}. \end{aligned}$$

The right star state can be obtained symmetrically.

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Shi, Y., Guo, Y. An Essential Seventh-Order Weighted Compact Adaptive Scheme for Hyperbolic Conservation Laws. J Sci Comput 89, 60 (2021). https://doi.org/10.1007/s10915-021-01664-z

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