Advertisement

Numerical Simulation of Hyperbolic Conservation Laws Using High Resolution Schemes with the Indulgence of Fuzzy Logic

  • Ruchika Lochab
  • Vivek KumarEmail author
Conference paper
  • 25 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11974)

Abstract

The aim of this paper is to solve numerically a class of problems on conservation laws, modelled by hyperbolic partial differential equations. In this paper, primary focus is over the idea of fuzzy logic-based operators for the simulation of problems related to hyperbolic conservation laws. Present approach considers a novel computational procedure which relies on using some operators from fuzzy logic to reconstruct several higher-order numerical methods known as the flux-limited methods. Further optimization of the flux limiters is discussed. The approach ensures better convergence of the approximation and preserves the basic properties of the solution of the problem under consideration. The new limiters are further applied to several real-life problems like the advection problem to demonstrate that the optimized schemes ensure better results. Simulation results are included wherever required.

Keywords

Conservation laws Flux limiters Fuzzy logic 

Notes

Acknowledgement

RL thanks the Delhi Technological University for the partial financial support to attend NUMTA 2019 and UGC for PhD fellowship.

References

  1. 1.
    Strikwerda, J.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2004)zbMATHGoogle Scholar
  2. 2.
    LeVeque, R.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  3. 3.
    Laney, C.: Computational Gasdynamics, 1st edn. Cambridge University Press, New York (1998)CrossRefGoogle Scholar
  4. 4.
    Hirsch, C.: Numerical Computation of Internal and External Flows. Elsevier (2007)Google Scholar
  5. 5.
    Toro, E.: Riemann Solvers and Numerical methods for Fluid Dynamics. Springer, Heidelberg (1999).  https://doi.org/10.1007/978-3-662-03915-1CrossRefzbMATHGoogle Scholar
  6. 6.
    Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic, Theory and Applications (1995)Google Scholar
  7. 7.
    Breuss, M., Dietrich, D.: On the optimization of flux limiters for hyperbolic conservation laws. Numer. Methods Part. Differ. Equ. 29, 884–896 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chin, T., Qi, X.: Genetic algorithms for learning the rule base of fuzzy logic controller. Fuzzy Sets Syst. 97, 1–7 (1998)CrossRefGoogle Scholar
  9. 9.
    Kumar, V., Srinivasan, B.: An adaptive mesh strategy for singularly perturbed convection diffusion problem. Appl. Math. Model. 39, 2081–2091 (2015)MathSciNetCrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Applied MathematicsDelhi Technological UniversityDelhiIndia

Personalised recommendations