Skip to main content
Log in

A van Leer-Type Numerical Scheme for the Model of a General Fluid Flow in a Nozzle with Variable Cross Section

Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

A van Leer-type numerical scheme for the model of a general fluid in a nozzle with variable cross section is presented. The model is nonconservative, making it hard for standard numerical schemes. Exact solutions of local Riemann problems are incorporated in the construction of this scheme. The scheme can work well in regions of resonance, where multiple waves are colliding. Numerical tests are conducted, where we compare the errors and orders of accuracy for approximating exact solutions between this scheme and a Godunov-type scheme. Results from numerical tests show that this van Leer-type scheme has a much better accuracy than the Godunov-type scheme.

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

Access this article

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

Similar content being viewed by others

References

  1. Ambroso, A., Chalons, C., Coquel, F., Galié, T.: Relaxation and numerical approximation of a two-fluid two-pressure diphasic model. Math. Mod. Numer. Anal. 43, 1063–1097 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambroso, A., Chalons, C., Raviart, P. -A.: A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput. Fluids 54, 67–91 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Audusse, E., Bouchut, F., Bristeau, M. -O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben-Artzi, M., Falcovitz, J.: Generalized Riemann Problems: from the Scalar Equation to Multidimensional Fluid Dynamics Recent Advances in Computational Sciences, vol. 1–49. World Sci. Publ., Hackensack (2008)

  5. Bermúdez, A., López, X. L., Vázquez-Cendón, M. E.: Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines. J. Comput. Phys. 323, 126–148 (2016)

    Article  MathSciNet  Google Scholar 

  6. Bernetti, R., Titarev, V. A., Toro, E. F.: Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry. J. Comput. Phys. 227, 3212–3243 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Frontiers in Mathematics Series, Birkhäuser (2004)

    Book  MATH  Google Scholar 

  8. Botchorishvili, R., Perthame, B., Vasseur, A.: Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comput. 72, 131–157 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Botchorishvili, R., Pironneau, O.: Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws. J. Comput. Phys. 187, 391–427 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castro, C. E., Toro, E. F.: A Riemann solver and upwind methods for a two-phase flow model in non-conservative form. Internat. J. Numer. Methods Fluids 50, 275–307 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cuong, D. H., Thanh, M. D.: A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section. Appl. Math. Comput. 256, 602–629 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Cuong, D. H., Thanh, M. D.: Building a Godunov-type numerical scheme for a model of two-phase flows. Comput. Fluids 148, 69–81 (2017)

    Article  MathSciNet  Google Scholar 

  13. Cuong, D.H., Thanh, M.D.: A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography. Adv. Comput. Math. https://doi.org/10.1007/s10444-017-9521-4

  14. Cuong, D. H., Thanh, M. D.: Constructing a Godunov-type scheme for the model of a general fluid flow in a nozzle with variable cross-section. Appl. Math. Comput. 305, 136–160 (2017)

    MathSciNet  Google Scholar 

  15. Cuong, D. H., Thanh, M. D.: A high-resolution van Leer-type scheme for a model of fluid flows in a nozzle with variable cross-section. J. Korean Math. Soc. 54(1), 141–175 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dal Maso, G., LeFloch, P. G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Godlewski, E., Raviart, P. A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York (1996)

    Book  MATH  Google Scholar 

  18. Hou, T. Y., LeFloch, P. G.: Why nonconservative schemes converge to wrong solutions. Error Anal. Math. Comput. 62, 497–530 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Isaacson, E., Temple, B.: Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52, 1260–1278 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Isaacson, E., Temple, B.: Convergence of the 22 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55, 625–640 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kröner, D., Thanh, M. D.: Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43, 796–824 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kröner, D., LeFloch, P. G., Thanh, M. D.: The minimum entropy principle for fluid flows in a nozzle with discntinuous crosssection. Math. Mod. Numer. Anal. 42, 425–442 (2008)

    Article  MATH  Google Scholar 

  23. LeFloch, P. G.: Shock waves for nonlinear hyperbolic systems in nonconservative form. Institute Math. Appl., Minneapolis. Preprint 593 (1989)

  24. LeFloch, P. G., Thanh, M. D.: The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Comm. Math. Sci. 1, 763–797 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. LeFloch, P. G., Thanh, M. D.: The Riemann problem for shallow water equations with discontinuous topography. Comm. Math. Sci. 5, 865–885 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. LeFloch, P. G., Thanh, M. D.: A Godunov-type method for the shallow water equations with variable topography in the resonant regime. J. Comput. Phys. 230, 7631–7660 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Marchesin, D., Paes-Leme, P. J.: A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differential equations III. Comput. Math. Appl. (Part A) 12, 433–455 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rosatti, G., Begnudelli, L.: The Riemann Problem for the one-dimensional, free-surface shallow water equations with a bed step: theoretical analysis and numerical simulations. J. Comput. Phys. 229, 760–787 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schwendeman, D. W., Wahle, C. W., Kapila, A. K.: The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212, 490–526 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Saurel, R., Abgrall, R.: A multi-phase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tian, B., Toro, E. F., Castro, C. E.: A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver. Comput. Fluids 46, 122–132 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Thanh, M. D.: The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area. SIAM J. Appl. Math. 69, 1501–1519 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Thanh, M. D.: A phase decomposition approach and the Riemann problem for a model of two-phase flows. J. Math. Anal. Appl. 418, 569–594 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Thanh, M. D., Cuong, D. H.: Existence of solutions to the Riemann problem for a model of two-phase flows. Elect. J. Diff. Eqs. 2015(32), 1–18 (2015)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the referee for his/her very constructive comments and fruitful discussions.

Funding

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.15.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mai Duc Thanh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thanh, M.D., Cuong, D.H. A van Leer-Type Numerical Scheme for the Model of a General Fluid Flow in a Nozzle with Variable Cross Section. Acta Math Vietnam 43, 503–547 (2018). https://doi.org/10.1007/s40306-017-0242-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-017-0242-z

Keywords

Mathematics Subject Classification (2010)

Navigation