Skip to main content
Springer Nature Link
Log in

Hybrid Optimized Low-Dissipation and Adaptive MUSCL Reconstruction Technique for Hyperbolic Conservation Laws

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A new hybrid optimized low-dissipation and adaptive MUSCL scheme is present for finite volume method. The proposed scheme, based on an optimized linear scheme with monotonicity preserving limitation and an adaptive MUSCL scheme, emphasizes on the resolution rather than the formal order. This technique is applied to cell interface reconstruction and bears similarity in form to the widely used MUSCL scheme except wider interpolation stencil. Although the scheme is not high order of accuracy because of adaptive MUSCL scheme acting as nonlinear part, the low-dissipation feature from the optimized linear part makes it very accurate and robust in many practical applications, where much richer flow structures can be obtained. A number of test cases are solved to validate the high resolution of the present scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Wang, Z.J., et al.: High-order CFD methods: current status and perspective. Int. J. Numer. Meth. Fluids 72, 811–845 (2013)

    Article  MathSciNet  Google Scholar 

  2. May, G., Jameson, A.: High-order accurate methods for high-speed flow. In: 17th AIAA Computational Fluid Dynamics Conference, 6–9 June 2005, Toronto, Ontario Canada, AIAA 2005-5251

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Sun, Z.-S., Ren, Y.-X.: A class of finite difference schemes with low dispersion and constrollable dissipation for DNS of compressible turbulence. J. Comput. Phys. 230, 4616–4635 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang, J., Li, Z., Lipeng, L.: An optimized low-dissipation monotonicity-preserving scheme for numerical simulations of high-speed turbulent flows. J. Sci. Comput. 56, 67–95 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, X.-l., Leng, Y., He, Z.-w.: Optimized sixth-order monotonicity-preserving scheme by nonlinear spectral analysis. Int. J. Numer. Meth. Fluids 73, 560–577 (2013)

    Article  MathSciNet  Google Scholar 

  10. Suresh, A., Huynh, H.T.: Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys. 136, 83–90 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Jammalamadaka, A., Li, Z., et al.: Subgrid-scale models for large-Eddy simulations of shock boundary layer interactions. AIAA J 51(5), 1174–1188 (2013)

    Article  Google Scholar 

  13. Sun, Z.-s., Luo, L., Ren, Y.-x., Zhang, S.-y.: A sixth order hybrid finite difference scheme based on the minimized dispersion and controllable dissipation technique. J. Comput. Phys. 270, 238–254 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. 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 

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

    Article  MATH  Google Scholar 

  20. QiuJu, W.A.N.G., YuXin, R.E.N., ZhenSheng, S.U.N., YuTao, S.U.N.: Low dispersion finite volume scheme based on reconstruction with minimized dispersion and controllable dissipation. Sci China Phys Mech Astron 56, 423–431 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Kim, D., Kwon, J.H.: A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis. J. Comput. Phys. 210, 554–583 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Billet, G., Louedin, O.: Adaptive limiters for improving the accuracy of the MUSCL approach for unsteady flows. J. Comput. Phys. 170, 161–183 (2001)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Spiteri, R.J., Ruuth, S.J.: Non-linear evolution using optimal fourth-order strong-stability-preserving Runge–Kutta methods. Math Comput Simul 62, 125–135 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Leveque, R.J.: Finite-volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  28. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3101–3211 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Fan, P., Shen, Y., Tian, B., Yang, C.: A new smoothness indicator for improving the weighted essentially non-oscillatory scheme. J. Comput. Phys. 269, 329–354 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toro, E.F., Titarev, V.A.: MUSTA fluxes for systems of conservation laws. J. Comput. Phys. 216, 403–429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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  MATH  Google Scholar 

  33. Shima, E., Kitamura, K.: Parameter-free simple low-dissipation AUSM-family scheme for all speeds. AIAA J 49(8), 1693–1709 (2011)

    Article  Google Scholar 

  34. Kitamura, K., Shima, E.: Towards shock-stable and accurate hypersonic heating computations: A new pressure flux for AUSM-family schemes. J. Comput. Phys. 245, 62–83 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Feng, H., Fuxing, H., Wang, R.: A new mapped weighted essentially non-oscillatory scheme. J. Sci. Comput. 51, 449–473 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer Method Partial Differ Equ 21(3), 536–552 (2005)

    Article  MATH  Google Scholar 

  37. Dumbser, M., et al.: ADER-WENO finite volume schemes with space-time adaptive mesh refinement. J. Comput. Phys. 248, 257–286 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Huang, K., et al.: Cures for numerical shock instability in HLLC solver. Int. J. Numer. Meth. Fluids 65, 1026–1038 (2011)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful for the valuable comments and suggestions made by the reviewers and my colleagues, which are significant contributions in improving the quality of this paper. Jie Wu was partially supported by the National Natural Science Foundation of China (91641107).

Author information

Authors and Affiliations

  1. School of Power and Energy, Northwestern Polytechnical University, Xi’an, 710072, China

    Jie Wu

  2. Science and Technology on Scramjet Laboratory, CARDC, Mianyang, 621000, China

    Jie Wu, Yuan-yuan He, Guo-hao Ding & Yi-yu Han

Authors
  1. Jie Wu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yuan-yuan He
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Guo-hao Ding
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Yi-yu Han
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jie Wu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, J., He, Yy., Ding, Gh. et al. Hybrid Optimized Low-Dissipation and Adaptive MUSCL Reconstruction Technique for Hyperbolic Conservation Laws. J Sci Comput 77, 552–578 (2018). https://doi.org/10.1007/s10915-018-0717-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0717-7

Keywords