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A New Hybrid WENO Scheme on a Four-Point Stencil for Euler Equations

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Abstract

To enhance the performance of the third-order weighted essentially non-oscillatory (WENO) scheme on the four-point stencil, a new hybrid central-type scheme is proposed in this paper. Developments are conducted in the following two aspects. First, a new fourth-order central WENO scheme, named as WENO4-LC, is devised for shock capturing, and a low dissipative third-order grid-centered upwind scheme is adopted for the smooth region. Second, a discontinuity indicator is extended to the four-point stencil version, which is simpler than its counterpart in our former article (Guo et al. in J Sci Comput 83, 28, 2020), for switching between the linear and nonlinear branch of the hybrid scheme. Numerical tests of the Euler and Navier–Stokes benchmarks show that the new indicator is good on detecting the discontinuities. Moreover, the performances of the central-type WENO4-LC and its corresponding hybrid scheme have obvious improvements compared with those of WENO3-JS and WENO3-L, and even behave slightly better than classical WENO5-JS on resolving the fluid structures. Meanwhile, the hybrid scheme is efficient which only costs 76% CPU time of WENO3-JS and about 58% of WENO5-JS.

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Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. Pirozzoli, S.: Numerical methods for high-speed flows. Ann. Rev. Fluid Mech. 43, 163–194 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Liu, S.P., Shen, Y.Q.: Discontinuity-detecting method for a four-point stencil and its application to develop a third-order hybrid-WENO scheme. J. Sci. Comput. 81, 1732–1766 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wu, X.S., Zhao, Y.X.: A high-resolution hybrid scheme for hyperbolic conservation laws. Int. J. Numer. Meth. Fluids 78, 162–187 (2015)

    Article  MathSciNet  Google Scholar 

  4. 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  MATH  Google Scholar 

  5. Zhou, J.Y., Adewumi, M.A.: Simulation of transients in natural gas pipelines using hybrid TVD schemes. Int. J. Numer. Meth. Fluids 32(4), 407–437 (2000)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Fu, L.: A hybrid method with TENO based discontinuity indicator for hyperbolic conservation laws. Commun. Comput. Phys. 26(4), 973–1007 (2019)

    Article  MathSciNet  Google Scholar 

  8. Li, G., Qiu, J.X.: Hybrid weighted essentially non-oscillatory schemes with different indicators. J. Comput. Phys. 229(21), 8105–8129 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhao, G.Y., Sun, M.B., Mei, Y., et al.: An efficient adaptive central-upwind WENO-CU6 numerical scheme with a new sensor. J. Sci. Comput. 81, 649–670 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Gande, N.R., Rathod, Y., Samala, R.: Improved third-order weighted essentially non-oscillatory scheme. Int. J. Numer. Methods Fluids 87, 329–342 (2018)

    Article  Google Scholar 

  15. Ha, Y.S., Kim, C.H., Yanng, H.S., et al.: Construction of an improved third-order WENO scheme with a new smoothness indicator. J. Sci. Comput. 82, 63 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wu, X.S., Liang, J.H., Zhao, Y.X.: A new smoothness indicator for third-order WENO scheme. Int. J. Numer. Methods Fluids 81, 451–459 (2016)

    Article  MathSciNet  Google Scholar 

  17. Li, C., Guo, Q.L., Sun, D., et al.: Improved third-order WENO schemes with new smoothness indicators. Int. J. Numer. Methods Fluids 93(1), 1–23 (2021)

    Article  Google Scholar 

  18. Martin, M.P., Taylor, E.M., Wu, M., et al.: A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220, 270–289 (2006)

    Article  MATH  Google Scholar 

  19. Li, Q., Guo, Q.L., Sun, D., et al.: A fourth-order symmetric WENO scheme with improved performance by new linear and nonlinear optimizations. J. Sci. Comput. 71, 109–143 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu, X.Y., Wang, Q., Adams, N.A.: An adaptive central-upwind weighted essentially non-oscillatory scheme. J. Comput. Phys. 229, 8952–8965 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hu, F.X.: The 6th-order weighted ENO schemes for hyperbolic conservation laws. Comput. Fluids 174, 34–45 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ducros, F., Ferrand, V., Nicoud, F., et al.: Large-eddy simulation of the shock/turbulence interaction. J. Comput. Phys. 152, 517–549 (1999)

    Article  MATH  Google Scholar 

  23. Li, Z., Ju, Y.P., Zhang, C.H.: Hybrid central–WENO scheme for the large eddy simulation of turbulent flows with shocks. Numer. Heat Transf. Part B 72(2), 170–189 (2017)

    Article  MathSciNet  Google Scholar 

  24. Movahed, P., Johnsen, E.: A solution-adaptive method for efficient compressible multifluid simulations, with application to the Richtmyer-Meshkov instability. J. Comput. Phys. 239, 166–186 (2013)

    Article  MathSciNet  Google Scholar 

  25. Hill, D.J., Pullin, D.I.: Hybrid tuned center-difference-WENO method for large-eddy simulations in the presence of strong shocks. J. Comput. Phys. 194, 435–450 (2004)

    Article  MATH  Google Scholar 

  26. Guo, Q.L., Sun, D., Li, C., et al.: A new discontinuity indicator for hybrid WENO schemes. J. Sci. Comput. 83, 28 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bhise, A.A., Gande, N.R., Samala, R., et al.: An efficient hybrid WENO scheme with a problem independent discontinuity locator. Int. J. Numer. Methods Fluids 91, 1–28 (2019)

    Article  MathSciNet  Google Scholar 

  28. Liu, S., Shen, Y., Chen, B., Zeng, F.: Novel local smoothness indicators for improving the third-order WENO scheme. Int. J. Numer. Methods Fluids 87(2), 51–69 (2017)

    Article  MathSciNet  Google Scholar 

  29. Steger, J.L., Warming, R.F.: Flux vector splitting of inviscid gasdynamics equations with application to finite difference methods. J. Comput. Phys. 40(2), 263–293 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pirozzoli, S.: On the spectral properties of shock-capturing schemes. J. Comput. Phys. 219, 489–497 (2006)

    Article  MATH  Google Scholar 

  31. Li, C., Guo, Q.L., Li, Q., et al.: Numerical simulations on hypersonic shock wave/boundary layer interactions by a third-order optimized symmetric WENO scheme. Trans. Nanjing Univ. Aeronaut. Astronaut. 34(5), 524–534 (2017)

    Google Scholar 

  32. Sun, D., Guo, Q.L., Li, C., et al.: Assessment of optimized symmetric fourth-order weighted essentially non-oscillatory scheme in direct numerical simulation of compressible turbulence. Comput. Fluids 197, 104383 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sun, D., Guo, Q.L., Li, C., et al.: Direct numerical simulation of effects of a micro-ramp on a hypersonic shock wave/boundary layer interaction. Phys. Fluids 31, 126101 (2019)

    Article  Google Scholar 

  34. Li, C., Guo, Q.L., Sun, D., et al.: Aerothermal prediction of hypersonic flow around spherical capsule model using IDDES approach. Int. J. Mod. Phys. B 34, 20400780 (2020)

    Article  Google Scholar 

  35. Jiang, Y., Shu, C.W., Zhang, M.P.: An alternative formulation of finite difference weighted ENO schemes with Lax-Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35(2), 1137–1160 (2015)

    Article  MathSciNet  Google Scholar 

  36. Lax, P.D.: Weak solution of non-linear hyperbolic equations and their numerical computations. Commun. Pure Appl. Math. 7, 159–193 (1954)

    Article  MATH  Google Scholar 

  37. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lax, P.D., Liu, X.D.: Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19, 319–340 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Young, Y.N., Tufo, H., Dubey, A., et al.: On the miscible Rayleigh-Taylor instability: two and three dimensions. J. Fluid Mech. 447, 377–408 (2001)

    Article  MATH  Google Scholar 

  41. Fu, L., Hu, X.Y., Adams, N.A.: A family of high-order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, S.H., Shu, C.W.: A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput. 31(1/2), 273–350 (2006)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Yee, H.C., Sandham, N.D., Djomehri, M.J.: Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150, 199–238 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liu, X.L., Zhang, S.H.: Direct numerical simulation of the interaction of 2D shock wave and shear layer. Chin. J. Theor. Appl. Mech. 45(1), 61–75 (2013). (in Chinese)

    Google Scholar 

  46. Samtaney, R., Pullin, D.I., Kosovic, B.: Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids 13(5), 1415–1430 (2001)

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Numerical Windtunnel project (NNW-NB-JC-027), the National Key Research and Development Program of China (2019YFA0405201), Fundamental and Frontier Technology Research Fund of CARDC (PJD20180204) and the National Natural Science Foundation of China (11802324, 12002360 and 92052301). The authors are thankful to the reviewers for their valuable suggestions.

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Authors and Affiliations

  1. State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, 621000, People’s Republic of China

    Chen Li, Dong Sun, Qilong Guo, Pengxin Liu & Hanxin Zhang

  2. Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, 621000, People’s Republic of China

    Chen Li, Dong Sun, Qilong Guo & Pengxin Liu

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  1. Chen Li
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Correspondence to Qilong Guo.

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Li, C., Sun, D., Guo, Q. et al. A New Hybrid WENO Scheme on a Four-Point Stencil for Euler Equations. J Sci Comput 87, 18 (2021). https://doi.org/10.1007/s10915-021-01424-z

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