Advertisement

A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions

  • J. Ridder
  • A. M. RufEmail author
Article
  • 25 Downloads

Abstract

We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky–Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this article also proves existence of entropy solutions for the initial-boundary value problem for the general Ostrovsky–Hunter equation. Additionally, we show uniqueness using Kružkov’s doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally.

Keywords

Ostrovsky–Hunter equation Short-pulse equation Vakhnenko equation Finite difference methods Monotone scheme Existence Uniqueness Stability Convergence Entropy solution Dirichlet boundary conditions 

Mathematics Subject Classification

65M06 35M33 35L35 

Notes

Acknowledgements

The authors want to thank Nils Henrik Risebro from the University of Oslo for many insightful discussions.

References

  1. 1.
    Amiranashvili, S., Vladimirov, A.G., Bandelow, U.: A model equation for ultrashort optical pulses around the zero dispersion frequency. Eur. Phys. J. D 58(2), 219–226 (2010)Google Scholar
  2. 2.
    Bardos, C., Leroux, A.Y., Nedelec, J.C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Boyd, J.P.: Ostrovsky and Hunter’s generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves). Eur. J. Appl. Math. 16(1), 65–81 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brunelli, J.C., Sakovich, S.: Hamiltonian structures for the Ostrovsky–Vakhnenko equation. Commun. Nonlinear Sci. Numer. Simul. 18(1), 56–62 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Coclite, G., Karlsen, K., Kwon, Y.S.: Initial-boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation. J. Funct. Anal. 257(12), 3823–3857 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Coclite, G., Ridder, J., Risebro, N.: A convergent finite difference scheme for the Ostrovsky–Hunter equation on a bounded domain. BIT Numer. Math. 57(1), 93–122 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Coclite, G., di Ruvo, L., Karlsen, K.: Some wellposedness results for the Ostrovsky–Hunter equation. In: Hyperbolic Conservation Laws and Related Analysis with Applications, pp. 143–159. Springer (2014)Google Scholar
  8. 8.
    Coclite, G.M., di Ruvo, L.: Oleinik type estimates for the Ostrovsky–Hunter equation. J. Math. Anal. Appl. 423(1), 162–190 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Coclite, G.M., di Ruvo, L.: Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky–Hunter equation. J. Hyperbolic Differ. Equ. 12(02), 221–248 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Coclite, G.M., di Ruvo, L.: Well-posedness results for the short pulse equation. Z. Angew. Math. Phys. 66(4), 1529–1557 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Coclite, G.M., di Ruvo, L.: Wellposedness of bounded solutions of the non-homogeneous initial boundary for the short pulse equation. Boll. Unione Mat. Italiana 8(1), 31–44 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Coclite, G.M., di Ruvo, L.: Well-posedness of the Ostrovsky–Hunter equation under the combined effects of dissipation and short-wave dispersion. J. Evol. Equ. 16(2), 365–389 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Coclite, G.M., di Ruvo, L., Karlsen, K.H.: The initial-boundary-value problem for an Ostrovsky–Hunter type equation. In: Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis pp. 97–109 (2018)Google Scholar
  14. 14.
    Dubois, F., Le Floch, P.: Boundary Conditions for Nonlinear Hyperbolic Systems of Conservation Laws, pp. 96–104. Vieweg+Teubner Verlag, Wiesbaden (1989)zbMATHGoogle Scholar
  15. 15.
    Grimshaw, R.H., Helfrich, K., Johnson, E.R.: The reduced Ostrovsky equation: integrability and breaking. Stud. Appl. Math. 129(4), 414–436 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws, vol. 152. Springer, New York (2015)zbMATHGoogle Scholar
  18. 18.
    Hunter, J.K.: Numerical solutions of some nonlinear dispersive wave equations. Lect. Appl. Math. 26, 301–316 (1990)MathSciNetGoogle Scholar
  19. 19.
    Karlsen, K., Risebro, N., Storrøsten, E.: \(L^1\) error estimates for difference approximations of degenerate convection–diffusion equations. Math. Comput. 83(290), 2717–2762 (2014)zbMATHGoogle Scholar
  20. 20.
    Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sb. 10(2), 217 (1970)Google Scholar
  21. 21.
    Kuznetsov, N.: Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys. 16(6), 105–119 (1976)Google Scholar
  22. 22.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the short-pulse equation. Dyn. PDE 6(4), 291–310 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the Ostrovsky–Hunter equation. SIAM J. Math. Anal. 42(5), 1967–1985 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    de Monvel, A.B., Shepelsky, D.: The Ostrovsky–Vakhnenko equation: a Riemann–Hilbert approach. C. R. Math. 352(3), 189–195 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Morrison, A., Parkes, E., Vakhnenko, V.: The N loop soliton solution of the Vakhnenko equation. Nonlinearity 12(5), 1427 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ohlberger, M., Vovelle, J.: Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method. Math. Comput. 75(253), 113–150 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ostrovsky, L.: Nonlinear internal waves in a rotating ocean. Oceanology 18(2), 119–125 (1978)Google Scholar
  28. 28.
    Parkes, E.: The stability of solutions of Vakhnenko’s equation. J. Phys. A Math. Gen. 26(22), 6469 (1993)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Parkes, E.: Explicit solutions of the reduced Ostrovsky equation. Chaos Solitons Fractals 31(3), 602–610 (2007)MathSciNetzbMATHGoogle Scholar
  30. 30.
    di Ruvo, L.: Discontinuous solutions for the Ostrovsky–Hunter equation and two-phase flows. Ph.D. thesis, University of Bari (2013)Google Scholar
  31. 31.
    Strub, S.I., Bayen, M.A.: Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling. Int. J. Robust Nonlinear Control 16(16), 733–748 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Schäfer, T., Wayne, C.: Propagation of ultra-short optical pulses in cubic nonlinear media. Physica D Nonlinear Phenom. 196(1), 90–105 (2004)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Stepanyants, Y.A.: On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons. Chaos Solitons Fractals 28(1), 193–204 (2006)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Vakhnenko, V.: Solitons in a nonlinear model medium. J. Phys. A Math. Gen. 25(15), 4181 (1992)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Vakhnenko, V., Parkes, E.: The two loop soliton solution of the Vakhnenko equation. Nonlinearity 11(6), 1457 (1998)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Vakhnenko, V., Parkes, E.: The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos Solitons Fractals 13(9), 1819–1826 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of MathematicsUniversity of OsloOsloNorway

Personalised recommendations