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Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

Published online by Cambridge University Press:  20 August 2015

Yan Xu  and
Chi-Wang Shu
Yan Xu*
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
Corresponding author.Email:yxu@ustc.edu.cn
Email:shu@dam.brown.edu
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Abstract

In this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions. The proof of the total variation stability of the schemes for the piecewise constant P0 case is also given. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the LDG method.

Keywords

65M6035Q53Local discontinuous Galerkin methodDegasperis-Procesi equationL2 stabilitytotal variation stability
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 474 - 508
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Bassi, F., and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267–279.CrossRefGoogle Scholar
[2] Cockburn, B., Hou, S., and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54 (1990), 545–581.Google Scholar
[3] Cockburn, B., Lin, S.-Y., and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. Comput. Phys., 84 (1989), 90–113.Google Scholar
[4] Cockburn, B., and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52 (1989), 411–435.Google Scholar
[5] Cockburn, B., and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440–2463.Google Scholar
[6] Cockburn, B., and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), 199–224.Google Scholar
[7] Cockburn, B., and Shu, C.-W., Foreword for the special issue on discontinuous Galerkin method, J. Sci. Comput., 22-23 (2005), 1–3.Google Scholar
[8] Cockburn, B., and Shu, C.-W., Foreword for the special issue on discontinuous Galerkin method, J. Sci. Comput., 40 (2009), 1–3.Google Scholar
[9] Coclite, G., and Karlsen, K., On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60–91.Google Scholar
[10] Coclite, G., and Karlsen, K., On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation, J. Differential Equations, 234 (2007), 142–160.Google Scholar
[11] Coclite, G., Karlsen, K., and Risebro, N., Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28 (2008), 8–105.Google Scholar
[12] Dawson, C., Foreword for the special issue on discontinuous Galerkin method, Comput. Meth. Appl. Mech. Eng., 195 (2006), 3183.Google Scholar
[13] Degasperis, A., Holm, D., and Hone, A., A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463–1474.Google Scholar
[14] Degasperis, A., and Procesi, M., Asymptotic integrability, in Degasperis, A. and Gaeta, G., editors, Symmetry and Perturbation Theory, 23–37, World Sci. Publ., River Edge, NJ, 1999.Google Scholar
[15] El-Shehawey, M., El-Shreef, G., and Al-Henawy, A., Analytical inversion of general periodic tridiagonal matrices, J. Math. Anal. Appl., 345 (2008), 123–134.Google Scholar
[16] Escher, J., Liu, Y., and Yin, Z., Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457–485.CrossRefGoogle Scholar
[17] Escher, J., Liu, Y., and Yin, Z., Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87–117.Google Scholar
[18] Feng, B., and Liu, Y., An operator splitting method for the Degasperis-Procesi equation, J. Comput. Phys., 228 (2009), 7805–7820.Google Scholar
[19] Hesthaven, J., and Warburton, T., Nodal Discontinuous Galerkin Methods, Algorithms, Analysis, and Applications, Springer, 2008.Google Scholar
[20] Hoel, H., A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation, Electron. J. Differential Equations, 2007 (2007), 1–22.Google Scholar
[21] Lenells, J., Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72–82.Google Scholar
[22] Li, B., Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Springer-Verlag, London, 2006.CrossRefGoogle Scholar
[23] Liu, Y., and Yin, Z., Global existence and blow-up phenomena for the Degasperis-Procesi equation, Commun. Math. Phys., 267 (2006), 801–820.Google Scholar
[24] Lundmark, H., Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear. Sci., 17 (2007), 169–198.Google Scholar
[25] Lundmark, H., and Szmigielski, J., Multi-peakon solutions of the Degasperis-Procesi equation, Inverse. Prob., 19 (2003), 1241–1245.CrossRefGoogle Scholar
[26] Lundmark, H., and Szmigielski, J., Degasperis-Procesi peakons and the discrete cubic string, IMRP Int. Math. Res. Pap., 2 (2005), 53–116.Google Scholar
[27] Matsuno, Y., Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit, Inverse. Prob., 21 (2005), 1553–1570.Google Scholar
[28] Matsuno, Y., The N-soliton solution of the Degasperis-Procesi equation, Inverse. Prob., 21 (2005), 2085–2101.Google Scholar
[29] Reed, W., and Hill, T., Triangular Mesh Methods for the Neutrontransport Equation, La-ur-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[30] Riviére, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, SIAM, 2008.Google Scholar
[31] Shu, C.-W., Discontinuous Galerkin methods: general approach and stability, in Bertoluzza, S., Falletta, S., Russo, G., and Shu, C.-W., editors, Numerical Solutions of Partial Differential Equations, 149–201, Birkhauser, Basel, 2009.Google Scholar
[32] Shu, C.-W., and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439–471.Google Scholar
[33] Xu, Y., and Shu, C.-W., Local discontinuous Galerkin methods for two classes of two dimensional nonlinear wave equations, Phys. D., 208 (2005), 21–58.CrossRefGoogle Scholar
[34] Xu, Y., and Shu, C.-W., A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal., 46 (2008), 1998–2021.Google Scholar
[35] Xu, Y., and Shu, C.-W., Local discontinuous Galerkin method for the Hunter-Saxton equation and its zero-viscosity and zero-dispersion limit, SIAM J. Sci. Comput., 31 (2008), 1249–1268.Google Scholar
[36] Xu, Y., and Shu, C.-W., Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7 (2010), 1–46.Google Scholar
[37] Xu, Y., and Shu, C.-W., Dissipative numerical methods for the Hunter-Saxton equation, J. Comput. Math., 28 (2010), 606–620.Google Scholar
[38] Yan, J., and Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), 769–791.Google Scholar
[39] Yin, Z., Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129–139.Google Scholar
[40] Yin, Z., On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649–666.Google Scholar
[41] Yin, Z., Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182–194.Google Scholar