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Entropy Stable Flux Correction for Scalar Hyperbolic Conservation Laws

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Abstract

It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the monotone scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution of scalar hyperbolic conservation laws. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. We present examples that show that not all numerical entropy fluxes guarantee to single out a physically correct solution of scalar hyperbolic conservation laws.

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Data Availability Statement

Enquiries about data availability should be directed to the authors.

References

  1. Boris, J.P., Book, D.L.: Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11(1), 38–69 (1973). https://doi.org/10.1016/0021-9991(73)90147-2

    Article  MATH  Google Scholar 

  2. Chalons, C., LeFloch, P.: A fully discrete scheme for diffusive-dispersive conservation laws. Numer. Math. 89(3), 493–509 (2001). https://doi.org/10.1007/PL00005476

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017). https://doi.org/10.1016/j.jcp.2017.05.025

    Article  MathSciNet  MATH  Google Scholar 

  4. Fjordholm, U., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012). https://doi.org/10.1137/110836961

    Article  MathSciNet  MATH  Google Scholar 

  5. Harten, A., Hyman, J.M., Lax, P.D., Keyfitz, B.: On finite-difference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29(3), 297–322 (1976). https://doi.org/10.1002/cpa.3160290305

    Article  MathSciNet  MATH  Google Scholar 

  6. Hiltebrand, A., Mishra, S.: Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws. Numer. Math. 126(1), 103–151 (2014). https://doi.org/10.1007/s00211-013-0558-0

    Article  MathSciNet  MATH  Google Scholar 

  7. Kivva, S.: Flux-corrected transport for scalar hyperbolic conservation laws and convection-diffusion equations by using linear programming. J. Comput. Phys. 425, 109874 (2021). https://doi.org/10.1016/j.jcp.2020.109874

    Article  MathSciNet  Google Scholar 

  8. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160(1), 241–282 (2000). https://doi.org/10.1006/jcph.2000.6459

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuzmin, D.: Explicit and implicit FEM-FCT algorithms with flux linearization. J. Comput. Phys. 228(7), 2517–2534 (2009). https://doi.org/10.1016/j.jcp.2008.12.011

    Article  MathSciNet  MATH  Google Scholar 

  10. Kuzmin, D., Möller, M.: Algebraic flux correction I. Scalar conservation laws. in Flux-Corrected Transport: Principles, Algorithms, and Applications, Berlin, Springer, 2005, pp. 155-206. https://doi.org/10.1007/3-540-27206-2_6

  11. Kuzmin, D., Moller, M., Turek, S.: High-resolution FEM-FCT schemes for multidimensional conservation laws. Comput. Methods Appl. Mech. Eng. 193(45–47), 4915–4946 (2004). https://doi.org/10.1016/j.cma.2004.05.009

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuzmin, D., Turek, S.: Flux correction tools for finite elements. J. Comput. Phys. 175(2), 525–558 (2002). https://doi.org/10.1006/jcph.2001.6955

    Article  MathSciNet  MATH  Google Scholar 

  13. Lax, P.: Hyperbolic Systems of Conservation Laws and Mathematical Theory of Shock Waves. In: Vol.11 of SIAM Regional Conference Series in Applied Mathematics (1972)

  14. Lefloch, P., Mercier, J.-M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002). https://doi.org/10.1137/S003614290240069X

    Article  MathSciNet  MATH  Google Scholar 

  15. LeVeque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33(2), 627–665 (1996). https://doi.org/10.1137/0733033

    Article  MathSciNet  MATH  Google Scholar 

  16. Merriam, M.L.: An entropy-based approach to nonlinear stability. NASA-TM-101086, Ames Research Center, Moffett Field, California (1989)

  17. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  18. Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984). https://doi.org/10.1137/0721016

    Article  MathSciNet  MATH  Google Scholar 

  19. Osher, S., Chakravarthy, S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21(5), 955–984 (1984). https://doi.org/10.1137/0721060

    Article  MathSciNet  MATH  Google Scholar 

  20. Rusanov, V.: The calculation of the interaction of non-stationary shock waves and obstacles. USSR Comput. Math. Math. Phys. 1(2), 304–320 (1962). https://doi.org/10.1016/0041-5553(62)90062-9

    Article  Google Scholar 

  21. Sonar, T.: Entropy production in second-order three-point schemes. Numer. Math. 62, 371–390 (1992). https://doi.org/10.1007/BF01396235

    Article  MathSciNet  MATH  Google Scholar 

  22. Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43(168), 369–381 (1984). https://doi.org/10.1090/S0025-5718-1984-0758189-X

    Article  MathSciNet  MATH  Google Scholar 

  23. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987). https://doi.org/10.2307/2008251

    Article  MathSciNet  MATH  Google Scholar 

  24. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156

    Article  MathSciNet  MATH  Google Scholar 

  25. Tadmor, E.: Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete Contin. Dyn. Syst. 36(8), 4579–4598 (2016). https://doi.org/10.3934/dcds.2016.36.4579

    Article  MathSciNet  MATH  Google Scholar 

  26. Zakerzadeh, H., Fjordholm, U.: High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA J. Numer. Anal. 36(2), 633–654 (2016). https://doi.org/10.1093/imanum/drv020

    Article  MathSciNet  MATH  Google Scholar 

  27. Zalesak, S.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335–362 (1979). https://doi.org/10.1016/0021-9991(79)90051-2

    Article  MathSciNet  MATH  Google Scholar 

  28. Zalesak, S.T.: The Design of Flux-Corrected Transport (FCT) Algorithms For Structured Grids. In: Flux-Corrected Transport. Principles, Algorithms, and Applications. Springer, Berlin, pp. 29–78 (2005). https://doi.org/10.1007/3-540-27206-2_2

  29. Zhao, N., Wu, H.M.: MUSCL type schemes and discrete entropy conditions. J. Comput. Math. 15(1), 72–80 (1997)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank anonymous reviewers for their detailed comments, which helped us to significantly improve the paper.

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  1. Institute of Mathematical Machines and System Problems, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    Sergii Kivva

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Kivva, S. Entropy Stable Flux Correction for Scalar Hyperbolic Conservation Laws. J Sci Comput 91, 10 (2022). https://doi.org/10.1007/s10915-022-01792-0

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