Advertisement

Local Discontinuous Galerkin Methods for the \(\mu \)-Camassa–Holm and \(\mu \)-Degasperis–Procesi Equations

  • Chao Zhang
  • Yan Xu
  • Yinhua XiaEmail author
Article
  • 54 Downloads

Abstract

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the \(\mu \)-Camassa–Holm (\(\mu \)CH) and \(\mu \)-Degasperis–Procesi (\(\mu \)DP) equations. The conservative schemes for both two equations can preserve discrete versions of their own first two Hamiltonian invariants, while the dissipative ones guarantee the corresponding stability. The error estimates of both LDG schemes for the \(\mu \)CH equation are given. Comparing with the error estimates for the Camassa–Holm equation, some important tools are used to handle the unexpected terms caused by its particular Hamiltonian invariants. Moreover, a priori error estimates of two LDG schemes for the \(\mu \)DP equation are also proven in detail. Numerical experiments for both equations in different circumstances are provided to illustrate the accuracy and capability of these schemes and give some comparisons about their performance on simulations.

Keywords

\(\mu \)-Camassa–Holm equation \(\mu \)-Degasperis–Procesi equation Local discontinuous Galerkin methods Hamiltonian invariants Error estimates 

Notes

References

  1. 1.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bona, J., Chen, H., Karakashian, O., Xing, Y.: Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation. Math. Comput. 82(283), 1401–1432 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, Berlin (2007)Google Scholar
  4. 4.
    Chen, R.M., Lenells, J., Liu, Y.: Stability of the \(\mu \)-Camassa–Holm peakons. J. Nonlinear Sci. 23(1), 97–112 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia (2002)zbMATHGoogle Scholar
  6. 6.
    Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Karakashian, O., Xing, Y.: A posteriori error estimates for conservative local discontinuous Galerkin methods for the generalized Korteweg–de Vries equation. Commun. Comput. Phys. 20(1), 250–278 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Khesin, B., Lenells, J., Misiołek, G.: Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342(3), 617–656 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lenells, J., Misiołek, G., Tiğlay, F.: Integrable evolution equations on spaces of tensor densities and their peakon solutions. Commun. Math. Phys. 299(1), 129–161 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Liu, H., Xing, Y.: An invariant preserving discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Sci. Comput. 38(4), A1919–A1934 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Meng, X., Shu, C.-W., Wu, B.: Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85(299), 1225–1261 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous galerkin methods with implicit–explicit time-marching for advection–diffusion problems. SIAM J. Numer. Anal. 53(1), 206–227 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Xia, Y.: Fourier spectral methods for Degasperis–Procesi equation with discontinuous solutions. J. Sci. Comput. 61(3), 584–603 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Xia, Y., Xu, Y.: Weighted essentially non-oscillatory schemes for Degasperis–Procesi equation with discontinuous solutions. Ann. Math. Sci. Appl. 2(2), 319–340 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations. Physica D Nonlinear Phenom. 208(1–2), 21–58 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196(37), 3805–3822 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xu, Y., Shu, C.-W.: A local discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Numer. Anal. 46(4), 1998–2021 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Xu, Y., Shu, C.-W.: Local discontinuous Galerkin method for the Hunter–Saxton equation and its zero-viscosity and zero-dispersion limits. SIAM J. Sci. Comput. 31(2), 1249–1268 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Xu, Y., Shu, C.-W.: Dissipative numerical methods for the Hunter–Saxton equation. J. Comput. Math. 28, 606–620 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7(1), 1 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for the Degasperis–Procesi equation. Commun. Comput. Phys. 10(2), 474–508 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42(2), 641–666 (2004)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Zhang, Q., Xia, Y.: Conservative and dissipative local discontinuous Galerkin methods for Korteweg–de Vries type equations. Commun. Comput. Phys. 25(3), 532–563 (2019)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

Personalised recommendations