Advertisement

BIT Numerical Mathematics

, Volume 59, Issue 2, pp 547–563 | Cite as

Mimetic properties of difference operators: product and chain rules as for functions of bounded variation and entropy stability of second derivatives

  • Hendrik RanochaEmail author
Article

Abstract

For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear systems in several space dimensions is an open problem, mimetic properties such as summation-by-parts as discrete analogue of integration-by-parts allow a direct transfer of some results and their proofs, e.g. stability for linear systems. In this article, discrete analogues of the generalised product and chain rules that apply to functions of bounded variation are considered. It is shown that such analogues hold for certain second order operators but are not possible for higher order approximations. Furthermore, entropy dissipation by second derivatives with varying coefficients is investigated, showing again the far stronger mimetic properties of second order approximations compared to higher order ones.

Keywords

Product rule Chain rule Bounded variation Summation-by-parts Entropy stability Order barrier 

Mathematics Subject Classification

65M06 65M70 65N06 65N35 

Notes

Acknowledgements

This work was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/14-1. The author would like to thank the anonymous reviewers for their helpful comments and valuable suggestions to improve this article.

References

  1. 1.
    Bianchini, S., Bressan, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161(1), 223–342 (2005).  https://doi.org/10.4007/annals.2005.161.223 MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994).  https://doi.org/10.1006/jcph.1994.1057 MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148(2), 341–365 (1999).  https://doi.org/10.1006/jcph.1998.6114 MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    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 MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2010).  https://doi.org/10.1007/978-3-642-04048-1 zbMATHCrossRefGoogle Scholar
  6. 6.
    Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures et Appl. 74(6), 483–548 (1995)MathSciNetzbMATHGoogle Scholar
  7. 7.
    De Lellis, C., Otto, F., Westdickenberg, M.: Minimal entropy conditions for Burgers equation. Q. Appl. Math. 62(4), 687–700 (2004).  https://doi.org/10.1090/qam/2104269 MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Taylor & Francis Group, LLC, Boca Raton (2015)zbMATHCrossRefGoogle Scholar
  9. 9.
    Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014).  https://doi.org/10.1016/j.compfluid.2014.02.016 MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518–557 (2013).  https://doi.org/10.1016/j.jcp.2013.06.014 MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Fjordholm, U.S., 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 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233–A1253 (2013).  https://doi.org/10.1137/120890144 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39–66 (2016).  https://doi.org/10.1016/j.jcp.2016.09.013 MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gassner, G.J., Winters, A.R., Kopriva, D.A.: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291–308 (2016).  https://doi.org/10.1016/j.amc.2015.07.014 MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-Dependent Problems and Difference Methods. Wiley, New York (2013)zbMATHCrossRefGoogle Scholar
  16. 16.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer-Verlet method. Acta Numer. (2003).  https://doi.org/10.1017/S0962492902000144
  17. 17.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006).  https://doi.org/10.1007/3-540-30666-8 zbMATHCrossRefGoogle Scholar
  18. 18.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (2008).  https://doi.org/10.1007/978-3-540-78862-1 zbMATHCrossRefGoogle Scholar
  19. 19.
    Harten, A.: On the nonlinearity of modern shock-capturing schemes. In: Chorin, A.J., Majda, A.J. (eds.) Wave Motion: Theory, Modelling, and Computation, Mathematical Sciences Research Institute Publications, vol. 7, pp. 147–201. Springer, New York (1987).  https://doi.org/10.1007/978-1-4613-9583-6_6 CrossRefGoogle Scholar
  20. 20.
    Kreiss, H.O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24(3), 199–215 (1972).  https://doi.org/10.1111/j.2153-3490.1972.tb01547.x MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kreiss, H.O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York (1974)CrossRefGoogle Scholar
  22. 22.
    Krupa, S.G., Vasseur, A.F.: Single entropy condition for Burgers equation via the relative entropy method (2017). arXiv:1709.05610 [math.AP]
  23. 23.
    LeFloch, P.G.: Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Birkhäuser, Basel (2002).  https://doi.org/10.1007/978-3-0348-8150-0 zbMATHCrossRefGoogle Scholar
  24. 24.
    LeFloch, P.G., 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 MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Lefloch, P.G., Tzavaras, A.E.: Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30(6), 1309–1342 (1999).  https://doi.org/10.1137/S0036141098341794 MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51(3), 650–682 (2012).  https://doi.org/10.1007/s10915-011-9525-z MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mattsson, K., Svärd, M., Nordström, J.: Stable and accurate artificial dissipation. J. Sci. Comput. 21(1), 57–79 (2004).  https://doi.org/10.1023/B:JOMP.0000027955.75872.3f MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Mishra, S., Svärd, M.: On stability of numerical schemes via frozen coefficients and the magnetic induction equations. BIT Numer. Math. 50(1), 85–108 (2010).  https://doi.org/10.1007/s10543-010-0249-5 MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Panov, E.Y.: Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy. Math. Notes 55(5), 517–525 (1994).  https://doi.org/10.1007/BF02110380 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci. Comput. (2017).  https://doi.org/10.1007/s10915-017-0618-1. arXiv:1701.02264 [math.NA]
  31. 31.
    Ranocha, H.: Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods. GEM Int. J. Geomath. 8(1), 85–133 (2017).  https://doi.org/10.1007/s13137-016-0089-9. arXiv:1609.08029 [math.NA]
  32. 32.
    Ranocha, H.: Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws. Ph.D. Thesis, TU Braunschweig (2018)Google Scholar
  33. 33.
    Ranocha, H.: Generalised summation-by-parts operators and variable coefficients. J. Comput. Phys. 362, 20–48 (2018).  https://doi.org/10.1016/j.jcp.2018.02.021. arXiv:1705.10541 [math.NA]MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Ranocha, H., Glaubitz, J., Öffner, P., Sonar, T.: Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators. Appl. Numer. Math. 128, 1–23 (2018).  https://doi.org/10.1016/j.apnum.2018.01.019. See also arXiv:1606.00995 [math.NA] and arXiv:1606.01056 [math.NA]MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016).  https://doi.org/10.1016/j.jcp.2016.02.009. arXiv:1511.02052 [math.NA]MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ranocha, H., Öffner, P., Sonar, T.: Extended skew-symmetric form for summation-by-parts operators and varying Jacobians. J. Comput. Phys. 342, 13–28 (2017).  https://doi.org/10.1016/j.jcp.2017.04.044. arXiv:1511.08408 [math.NA]MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Raymond, J.P.: A new definition of nonconservative products and weak stability results. Bolletino Della Unione Matematica Italiana 10–B(7), 681–699 (1996)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence, RI (1997)zbMATHGoogle Scholar
  39. 39.
    Sjögreen, B., Yee, H.C.: On skew-symmetric splitting and entropy conservation schemes for the Euler equations. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds.) Numerical Mathematics and Advanced Applications 2009: Proceedings of ENUMATH 2009, the 8th European Conference on Numerical Mathematics and Advanced Applications, Uppsala, July 2009, pp. 817–827. Springer, Berlin (2010).  https://doi.org/10.1007/978-3-642-11795-4_88
  40. 40.
    Strand, B.: Summation by parts for finite difference approximations for \(d/dx\). J. Comput. Phys. 110(1), 47–67 (1994).  https://doi.org/10.1006/jcph.1994.1005 MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Svärd, M., Mishra, S.: Shock capturing artificial dissipation for high-order finite difference schemes. J. Sci. Comput. 39(3), 454–484 (2009).  https://doi.org/10.1007/s00033-012-0216-x MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014).  https://doi.org/10.1016/j.jcp.2014.02.031 MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    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/10.1090/S0025-5718-1987-0890255-3 MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    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 MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Vol’pert, A.I.: The spaces \(BV\) and quasilinear equations. Math. USSR-Sbornik 2(2), 225–267 (1967).  https://doi.org/10.1070/SM1967v002n02ABEH002340 zbMATHCrossRefGoogle Scholar
  46. 46.
    Vol’pert, A.I., Hudjaev, S.I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus Nijhoff Publishers, Dodrecht (1985)Google Scholar
  47. 47.
    von Neumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21(3), 232–237 (1950).  https://doi.org/10.1063/1.1699639 MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Wintermeyer, N., Winters, A.R., Gassner, G.J., Kopriva, D.A.: An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. J. Comput. Phys. 340, 200–242 (2017).  https://doi.org/10.1016/j.jcp.2017.03.036 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.TU BraunschweigBrunswickGermany

Personalised recommendations