SpringerLink
Log in

Efficient Implementation of the Hybrid Large Particle Method

  • Published:
Mathematical Models and Computer Simulations Aims and scope

Abstract

Within the framework of the hybrid method of large particles, an efficient algorithm with controlled numerical dissipation of the second order of accuracy in space and time is proposed. The computational properties of the algorithm are tested on one-dimensional Einfeldt, Tang and Liu, LeBlanc, Shu and Osher test problems, as well as two-dimensional Riemann problems. The algorithm is robust and has a high resolution, comparable to modern schemes, which formally have a higher order of approximation.

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
EUR 32.99 /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.

Similar content being viewed by others

REFERENCES

  1. S. K. Godunov, “A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics,” Sb. Math. 47 (8–9), 357–393 (1959).

    Google Scholar 

  2. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (Springer, Berlin, 2009). https://doi.org/10.1007/b79761

    Book  MATH  Google Scholar 

  3. A. Harten, “High resolution schemes for hyperbolic conservation laws,” J. Comput. Phys. 49 (3), 357–393 (1983). https://doi.org/10.1016/0021-9991(83)90136-5

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Hirsch, Numerical Computation of Internal and External Flows, Vol. 2: Computational Methods for Inviscid and Viscous Flows (Wiley, New York, 1990).

  5. X.-D. Liu, S. Osher, and T. Chan, “Weighted essentially non-oscillatory schemes,” J. Comput. Phys. 115 (1), 200–212 (1994). https://doi.org/10.1006/jcph.1994.1187

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Shi, Y.-T. Zhang, and C.-W. Shu, “Resolution of high order WENO schemes for complicated flow structures,” J. Comput. Phys. 186 (2), 690–696 (2003). https://doi.org/10.1016/S0021-9991(03)00094-9

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Cockburn and C.-W. Shu, “Runge–Kutta discontinuous Galerkin methods for convection-dominated problems,” J. Sci. Comput. 16 (3), 173–261 (2001). https://doi.org/10.1023/A:1012873910884

    Article  MathSciNet  MATH  Google Scholar 

  8. M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Application of the RKDG method for gas dynamics problems,” Math. Models Comput. Simul. 6 (4), 397–407 (2014). https://doi.org/10.1134/S207004821404005X

    Article  MathSciNet  MATH  Google Scholar 

  9. R. P. Fedorenko, “The application of difference schemes of high accuracy to the numerical solution of hyperbolic equations,” USSR Comput. Math. Math. Phys. 2 (6), 1355–1365 (1963). https://doi.org/10.1016/0041-5553(63)90351-3

    Article  MATH  Google Scholar 

  10. X. Liu, S. Zhang, H. Zhang, and C.-W. Shu, “A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes,” J. Comput. Phys. 284, 133–154 (2015). https://doi.org/10.1016/j.jcp.2014.12.027

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. A. Kriksin and V. F. Tishkin, “Hybrid approach to solve single-dimensional gas dynamics equations,” Math. Models Comput. Simul. 11 (2), 256–265 (2019). https://doi.org/10.1134/S2070048219020078

    Article  MathSciNet  Google Scholar 

  12. B. N. Chetverushkin, “Hyperbolic quasi-gasdynamic system,” Math. Models Comput. Simul. 10 (5), 588–600 (2018). https://doi.org/10.1134/S2070048218050046

    Article  MathSciNet  Google Scholar 

  13. R. Liska and B. Wendroff, “Comparison of several difference schemes on 1D and 2D test problems for the Euler equations,” SIAM J. Sci. Comput. 25 (3), 995–1017 (2003). https://doi.org/10.1137/S1064827502402120

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Tang and T. Liu, “A note on the conservative schemes for the Euler equations,” J. Comput. Phys. 218 (2), 451–459 (2006). https://doi.org/10.1016/j.jcp.2006.03.035

    Article  MATH  Google Scholar 

  15. A. V. Danilin and A.V. Solov’ev, “A modification of the CABARET scheme for resolving the sound points in gas flows,” Vychisl. Metody Program. 20 (4), 481–488 (2019). https://doi.org/10.26089/NumMet.v20r442

  16. I. Yu. Tagirova and A. V. Rodionov, “Application of artificial viscosity for suppressing the carbuncle phenomenon in Godunov-type schemes,” Math. Models Comput. Simul. 8 (3), 249–262 (2016). https://doi.org/10.1134/S2070048216030091

    Article  MathSciNet  MATH  Google Scholar 

  17. D. V. Sadin, “TVD scheme for stiff problems of wave dynamics of heterogeneous media of nonhyperbolic nonconservative type,” Comput. Math. Math. Phys. 56 (12), 2068–2078 (2016). https://doi.org/10.1134/S0965542516120137

    Article  MathSciNet  MATH  Google Scholar 

  18. D. V. Sadin, “Schemes with customizable dissipative properties as applied to gas-suspensions flow simulation,” Mat. Model. 29 (12), 89–104 (2017).

    MathSciNet  Google Scholar 

  19. D. V. Sadin, “Application of scheme with customizable dissipative properties for gas flow calculation with interface instability evolution,” Nauchno-Tekh. Vestn. Inf. Tekhnol., Mekh. Opt. 8 (1), 153–157 (2018). https://doi.org/10.17586/2226-1494-2018-18-1-153-157

    Article  Google Scholar 

  20. D. V. Sadin, “A modification of the large-particle method to a scheme having the second order of accuracy in space and time for shockwave flows in a gas suspension,” Vestn. Yuzhno-Ural. Gos. Univ. Ser. Mat. Model. Program. 12 (2), 112–122 (2019). https://doi.org/10.14529/mmp190209

    Article  MATH  Google Scholar 

  21. D. V. Sadin and V. A. Davidchuk, “Comparison of a modified large-particle method with some high resolution schemes. One-dimensional test problems,” Vychisl. Metody Program. 20 (2), 138–146 (2019). https://doi.org/10.26089/NumMet.v20r214

  22. D. V. Sadin, I. O. Golikov, and E. N. Shirokova, “Testing of the hybrid large-particle method using two-dimensional Riemann problems,” St. Petersburg Polytech. State Univ. J. Phys. Math. 14 (1), 55–68 (2021). https://doi.org/10.18721/JPM.14104

    Article  Google Scholar 

  23. D. V. Sadin, “Analysis of dissipative properties of a hybrid large-particle method for structurally complicated gas flows,” Komp. Issled. Model. 12 (4), 757–772 (2020). https://doi.org/10.20537/2076-7633-2020-12-4-757-772

  24. M. E. Ladonkina, O. A. Neklyudova, and V. F. Tishkin, “Constructing a limiter based on averaging the solutions for the discontinuous Galerkin method,” Math Models Comput. Simul. 11 (1), 61–73 (2019). https://doi.org/10.1134/S2070048219010101

    Article  MathSciNet  Google Scholar 

  25. S. A. Karabasov, “On the power of second-order accurate numerical methods for model problems of gas- and hydrodynamics,” Math. Models Comput. Simul. 3 (1), 92–112 (2011). https://doi.org/10.1134/S2070048211010078

    Article  Google Scholar 

  26. Z. He, Y. Zhang, F. Gao, X. Li, and B. Tian, “An improved accurate monotonicity-preserving scheme for the Euler equations,” Comput. Fluids 140, 1–10 (2016). https://doi.org/10.1016/j.compfluid.2016.09.002

    Article  MathSciNet  MATH  Google Scholar 

  27. J. A. Greenough and W. J. Rider, “A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov,” J. Comput. Phys. 196 (1), 259–281 (2004). https://doi.org/10.1016/j.jcp.2003.11.002

    Article  MathSciNet  MATH  Google Scholar 

  28. X. Deng, B. **e, R. Loubère, Y. Shimizu, and F. **ao, “Limiter-free discontinuity-capturing scheme for compressible gas dynamics with reactive fronts,” Comput. Fluids 171, 1–14 (2018). https://doi.org/10.1016/j.compfluid.2018.05.015

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mozhaisky Military Space Academy, St. Petersburg, Russia

    D. V. Sadin

Authors
  1. D. V. Sadin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. V. Sadin.

Ethics declarations

The author declares that he has no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sadin, D.V. Efficient Implementation of the Hybrid Large Particle Method. Math Models Comput Simul 14, 946–954 (2022). https://doi.org/10.1134/S207004822206014X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S207004822206014X

Keywords:

Search

Navigation