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High Resolution TVD Scheme Based on Fuzzy Modifiers for Shallow-Water Equations

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Computational Science – ICCS 2021 (ICCS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12742))

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Abstract

This work proposes a new fuzzy logic based high resolution (HR), total variation diminishing (TVD) scheme in finite volume frameworks to compute an approximate solution of the shallow water equations (SWEs). Fuzzy logic enhances the execution of classical numerical algorithms. To test the effectiveness and accuracy of the proposed scheme, the dam-break problem is considered. A comparison of the numerical results by implementing some classical flux limiting methods is provided. The proposed scheme is able to capture both smooth and discontinuous profiles, leading to better oscillation-free results.

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References

  1. LeVeque, R.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  2. Xu, J., Liao, Z., Hu, Z.: A class of linear differential dynamical systems with fuzzy initial condition. Fuzzy Sets Syst. 158(21), 2339–2358 (2007)

    Article  MathSciNet  Google Scholar 

  3. Belotserkovskii, O.M.: Methods of computational gasdynamics. Computational Gasdynamics. ICMS, vol. 40, pp. 219–407. Springer, Vienna (1975). https://doi.org/10.1007/978-3-7091-2732-2_2

    Chapter  MATH  Google Scholar 

  4. Hirsch, C.: Numerical Computation of Internal and External Flows. Elsevier, Amsterdam (2007)

    Google Scholar 

  5. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall PTR, Upper Saddle River (1995)

    Google Scholar 

  6. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  Google Scholar 

  7. Breuss, M., Dietrich, D.: On the optimization of flux limiters for hyperbolic conservation laws. Numer. Methods Partial. Differ. Equ. 29 884–896 (2013)

    Google Scholar 

  8. Chin, T., Qi, X.: Genetic Algorithms for learning the rule base of fuzzy logic controller. Fuzzy Sets Syst. 97, 1–7 (1998)

    Google Scholar 

  9. Kumar, V., Srinivasan, B.: An adaptive mesh strategy for singularly perturbed convection diffusion problem. Appl. Math. Model. 39, 2081–2091 (2015)

    Article  MathSciNet  Google Scholar 

  10. Kumar, V., Rao, R.: Composite scheme using localized relaxation non-standard finite difference method for hyperbolic conservation laws. J. Sound Vib. 311, 786–801 (2008)

    Article  Google Scholar 

  11. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009)

    Google Scholar 

  12. Vázquez-Cendón, M.E.: Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148(2), 497–526 (1999)

    Article  MathSciNet  Google Scholar 

  13. Vreugdenhil, C.B.: Numerical Methods for Shallow-Water Flow. Springer, Netherlands (1994)

    Google Scholar 

  14. Kurganov, A.: Finite-volume schemes for shallow-water equations. Acta Numer. 27, 289–351 (2018)

    Article  MathSciNet  Google Scholar 

  15. Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: the Finite Volume Method. \(2^{nd}\)ed. Pearson Education Ltd, Gosport, Hants (2007)

    Google Scholar 

  16. Li, J., Du, Z.: A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. Hyperbolic conservation laws, SIAM J. Sci. Comput. 38(5), A3046–A3069 (2016)

    Google Scholar 

  17. Sergeyev, Y.D., Kvasov, D.E. (eds.): NUMTA 2019. LNCS, vol. 11973. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39081-5

    Book  Google Scholar 

  18. Lochab, R., Kumar, V.: An improved flux limiter using fuzzy modifiers for hyperbolic conservation laws. Math. Comput. Simulation 181, 16–37 (2021)

    Article  MathSciNet  Google Scholar 

  19. Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5), 995–1011 (1984)

    Article  MathSciNet  Google Scholar 

  20. Roe, P.L.: Characteristic-based schemes for the Euler equations. Ann. Rev. fluid Mech. 18, 337–365 (1986)

    Google Scholar 

  21. Van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14(4), 361–370 (1974)

    Google Scholar 

  22. Lochab, R., Kumar, V.: A new reconstruction of numerical fluxes for conservation laws using fuzzy operators. Int. J. Numer. Meth. Fluids. 93, 1690–1711 (2021). https://doi.org/10.1002/fld.4948

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Acknowledgement

The authors would like to express their gratitude to the anonymous reviewers for their keen observations and insightful comments.

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Authors and Affiliations

  1. Department of Applied Mathematics, Delhi Technological University, Delhi, 110042, India

    Ruchika Lochab & Vivek Kumar

Authors
  1. Ruchika Lochab
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  2. Vivek Kumar
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Corresponding author

Correspondence to Vivek Kumar .

Editor information

Editors and Affiliations

  1. AGH University of Science and Technology, Krakow, Poland

    Maciej Paszynski

  2. Ludwig-Maximilians-Universität München, Munich, Germany

    Dieter Kranzlmüller

  3. University of Amsterdam, Amsterdam, The Netherlands

    Valeria V. Krzhizhanovskaya

  4. University of Tennessee at Knoxville, Knoxville, TN, USA

    Jack J. Dongarra

  5. University of Amsterdam, Amsterdam, The Netherlands

    Peter M. A. Sloot

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Lochab, R., Kumar, V. (2021). High Resolution TVD Scheme Based on Fuzzy Modifiers for Shallow-Water Equations. In: Paszynski, M., Kranzlmüller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2021. ICCS 2021. Lecture Notes in Computer Science(), vol 12742. Springer, Cham. https://doi.org/10.1007/978-3-030-77961-0_39

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  • DOI: https://doi.org/10.1007/978-3-030-77961-0_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-77960-3

  • Online ISBN: 978-3-030-77961-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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