Skip to main content
SpringerLink
Log in

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

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

Abstract

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge–Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. Adams, N., 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 

  2. Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85, 257–283 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Castro, M., Costa, B., Don, W.: 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 

  5. Cockburn, B., Shu, C.-W.: Nonlinearly stable compact schemes for shock calculations. SIAM J. Numer. Anal. 31, 607–627 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dafermos, C.: Conservation laws in continuum physics, 3rd edn. Springer-Verlag, Berlin, Heidelberg (2010)

  7. Deng, X., Maekawa, H.: Compact high-order accurate nonlinear schemes. J. Comput. Phys. 130, 77–91 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. E, W., Liu, J.-G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122–138 (1996)

  10. Ekaterinaris, J.A.: Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics. J. Comput. Phys. 156, 272–299 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gaitonde, D., Shang, J.: Optimized compact-difference-based finite-volume schemes for linear wave phenomena. J. Comput. Phys. 138, 617–643 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghadimi, M., Farshchi, M.: Fourth order compact finite volume scheme on nonuniform grids with multi-blocking. Comput. Fluids 56, 1–16 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Ghosh, D., Baeder, J.D.: Weighted non-linear compact schemes for the direct numerical simulation of compressible, turbulent flows. J. Sci. Comput. 61, 61–89 (2014)

  15. Ghosh, D., Medida, S., Baeder, J.D.: Application of compact-reconstruction weighted essentially nonoscillatory schemes to compressible aerodynamic flows. AIAA J. 52, 1858–1870 (2014)

  16. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Hokpunna, A., Manhart, M.: Compact fourth-order finite volume method for numerical solutions of Navier–Stokes equations on staggered grids. J. Comput. Phys. 229, 7545–7570 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Kobayashi, M.: On a class of Padé finite volume methods. J. Comput. Phys. 156, 137–180 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Lerat, A., Corre, C.: A residual-based compact scheme for the compressible Navier–Stokes equations. J. Comput. Phys. 170, 642–675 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. LeVeque, R.J.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liang, C., Xu, Z.: Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws. J. Sci. Comput. 58, 41–60 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, J.-G., Shu, C.-W.: A high-order discontinuous Galerkin method for 2D incompressible flows. J. Comput. Phys. 160, 577–596 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, X., Zhang, S., Zhang, H., Shu, C.-W.: A new class of central compact schemes with spectral-like resolution I: linear schemes. J. Comput. Phys. 248, 235–256 (2013)

    Article  MathSciNet  Google Scholar 

  31. Osher, S., Chakravarthy, S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21, 955–984 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  32. Park, J.S., Yoon, S.-H., Kim, C.: Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys. 229, 788–812 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pereira, J., Kobayashi, M., Pereira, J.: A fourth-order-accurate finite volume compact method for the incompressible Navier–Stokes solutions. J. Comput. Phys. 167, 217–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Piller, M., Stalio, E.: Finite-volume compact schemes on staggered grids. J. Comput. Phys. 197, 299–340 (2004)

    Article  MATH  Google Scholar 

  35. Piller, M., Stalio, E.: Compact finite volume schemes on boundary-fitted grids. J. Comput. Phys. 227, 4736–4762 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Qiu, J.-M., Shu, C.-W.: Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys. 230, 863–889 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ren, Y., 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 

  39. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wilson, R.V., Demuren, A.O., Carpenter, M.: Higher-order compact schemes for numerical simulation of incompressible flows. ICASE Report 98-13 (1998)

  41. Xiong, T., Qiu, J.-M., Xu, Z.: A parametrized maximum principle preserving flux limiter for finite difference RK–WENO schemes with applications in incompressible flows. J. Comput. Phys. 252, 310–331 (2013)

    Article  MathSciNet  Google Scholar 

  42. Xu, Z.: Parametrized maximum principle preserving flux limiters for high order scheme solving hyperbolic conservation laws: one-dimensional scalar problem. Math. Comput. 83, 2213–2238 (2014)

    Article  MATH  Google Scholar 

  43. Yamaleev, N., Carpenter, M.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228, 4248–4272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, S., Jiang, S., 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 

  45. Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhang, X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 2752–2776 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The work was partly supported by the Fundamental Research Funds for the Central Universities (2010QNA39, 2010LKSX02). The third author acknowledges the funding support of this research by the Fundamental Research Funds for the Central Universities (2012QNB07).

Author information

Authors and Affiliations

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

    Yan Guo

  2. Department of Mathematics, University of Houston, Houston, TX, 77004, USA

    Tao Xiong

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

    Yufeng Shi

Authors
  1. Yan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yufeng Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Xiong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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, 83–109 (2015). https://doi.org/10.1007/s10915-014-9954-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9954-6

Keywords

Search

Navigation