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On Invariant-Preserving Finite Difference Schemes for the Camassa-Holm Equation and the Two-Component Camassa-Holm System

Part of: Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 April 2016

Hailiang Liu  and
Terrance Pendleton
Hailiang Liu*
Affiliation:
Iowa State University, Mathematics Department, Ames, IA 50011, USA
Terrance Pendleton
Affiliation:
Iowa State University, Mathematics Department, Ames, IA 50011, USA
*
*Corresponding author. Email addresses:[email protected] (H. Liu), [email protected] (T. Pendleton)
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Abstract

The purpose of this paper is to develop and test novel invariant-preserving finite difference schemes for both the Camassa-Holm (CH) equation and one of its 2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admitting soliton-like peakon solutions which are characterized by a slope discontinuity at the peak in the wave shape, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical schemes are shown to preserve two invariants, momentum and energy, hence numerically producing wave solutions with smaller phase error over a long time period than those generated by other conventional methods. We first apply the scheme to the CH equation and showcase the merits of considering such a scheme under a wide class of initial data. We then generalize this scheme to the 2CH equation and test this scheme under several types of initial data.

Keywords

Camassa-Holm equationPeakon solutionsenergy-preserving methods

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M12: Stability and convergence of numerical methods 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 1015 - 1041
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Artebrant, R. and Schroll, H., Numerical simulation of Camassa-Holm peakons by adaptive upwinding, Appl. Numer. Math., 56 (2006), pp. 695–711.Google Scholar
[2]Bressan, A., Chen, G., and Zhang, Q., Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discr. Cont. Dynam. Syst. 35 (2015), pp. 25–42.Google Scholar
[3]Bressan, A. and Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), pp. 215239.Google Scholar
[4]Camassa, R. and Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), pp. 1661–1664.Google Scholar
[5]Camassa, R. and Lee, L., A completely integrable particle method for a nonlinear shallow-water wave equation in periodic domains, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), pp. 1–5.Google Scholar
[6]Chertock, A., Du Toit, P., and Marsden, J. E., Integration of the EPDiff equation by particle methods, M2AN Math. Model. Numer. Anal., 46(3) (2012), pp. 515–534.CrossRefGoogle Scholar
[7]Chertock, A., Liu, J.-G., and Pendleton, T., Elastic collisions amongpeakon solutions for the Camassa-Holm equation, Appl. Numer. Math., 93 (2015), pp. 30–46.Google Scholar
[8]Chiu, P. H., Lee, L., and Sheu, T. W. H., A dispersion-relation-preserving algorithm for a non-linear shallow-water wave equation, J. Comput. Phys., 228 (2009), pp. 8034–8052.CrossRefGoogle Scholar
[9]Cohen, D., Matsuo, T., and Raynaud, X., A multi-symplectic numerical integrator for the two-component Camassa-Holm equation, J. Nonlinear Math. Phys., 21 (2014), pp. 442–453.Google Scholar
[10]Cohen, D., Owren, B., and Raynaud, X., Multi-symplectic integration of the Camassa-Holm equation, J. Comput. Phys., 227 (2008), pp. 5492–5512.Google Scholar
[11]Constantin, A. and Escher, J., Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), pp. 303–328.Google Scholar
[12]Constantin, A. and Ivanov, R. I., On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), pp. 7129–7132.CrossRefGoogle Scholar
[13]Constantin, A. and Kolev, B., On the geometric approach to the motion of inertial mechanical systems, J. Phys. A: Math. Gen., 35 (2002), pp. R51-R79.Google Scholar
[14]Constantin, A. and Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), pp. 787–804.CrossRefGoogle Scholar
[15]Escher, J., Lechtenfeld, O., and Yin, Z., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), pp. 493–513.Google Scholar
[16]Feng, B.-F., Maruno, K.-I., and Ohta, Y., A self-adaptive moving mesh method for the Camassa-Holm equation, J. Comput. Appl. Math., 235 (2010), pp. 229–243.CrossRefGoogle Scholar
[17]Gottlieb, S., Shu, C.-W., and Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89–112.Google Scholar
[18]Green, A. and Naghdi, P., A derivation of equations for wave propagation in water at variable depth, J. Fluid Mech., 78 (1976), pp. 237–246.Google Scholar
[19]Grunert, K., Blow-up for the two-component Camassa-Holm system., Discrete Contin. Dyn. Syst., 35(5) (2015), pp. 2041–2051.CrossRefGoogle Scholar
[20]Grunert, K., Holden, H., and Raynaud, X., Periodic conservative solutions for the two-component Camassa-Holm system, in Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th birthday, vol. 87 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2013, pp. 165–182.Google Scholar
[21]Grunert, K., Holden, H. and Raynaud, X., A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system, Forum Math., Sigma, 3 (2015), e1, doi:10.1017/fms.2014.29.CrossRefGoogle Scholar
[22]Holden, H. and Raynaud, X., A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), pp. 505–523.CrossRefGoogle Scholar
[23]Holm, D., Schmah, T., and Stoica, C., Geometric Mechanics and Symmetry, vol. 12 of Oxford Texts in Applied and Engineering Mathematics, Oxford University Press, Oxford, 2009.Google Scholar
[24]Holm, D. D., Náraigh, L. Ó, and Tronci, C., Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E (3), 79 (2009), pp. 016601, 13.Google Scholar
[25]Holm, D.-D. and Ivanov, R.-I, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, 43 (2010).Google Scholar
[26]Ivanov, R., Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), pp. 389–396.CrossRefGoogle Scholar
[27]Kalisch, H. and Lenells, J., Numerical study of traveling-wave solutions for the Camassa-Holm equation, Chaos Solitons Fractals, 25 (2005), pp. 287–298.CrossRefGoogle Scholar
[28]Kalisch, H. and Raynaud, X., Convergence of a spectral projection of the Camassa-Holm equation, Numer. Methods Partial Differential Equations, 22 (2006), pp. 1197–1215.Google Scholar
[29]Wang, Y., Song, Y., and Karimi, H. R., On the global dissipative and multipeakon dissipative behavior of the two-component Camassa-Holm system, Abstr. Appl. Anal., Article ID 348695 (2014), 16 pages.Google Scholar
[30]Kohlmann, M., The two-component Camassa-Holm system in weighted L p spaces, Z. Angew. Math. Mech. 94(3) (2014), pp. 264–272.Google Scholar
[31]Kurganov, A., Noelle, S., and Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23 (2001), pp. 707–740.Google Scholar
[32]Li, J. B. and Li, Y. S., Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation, Acta Math. Sin. (Engl. Ser.), 24 (2008), pp. 1319–1330.CrossRefGoogle Scholar
[33]Li, Y. and Olver, P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), pp. 27–63.Google Scholar
[34]Matsuo, T. and Yamaguchi, H., An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations, J. Comput. Phys., 228 (2009), pp. 4346–4358.Google Scholar
[35]Moon, B., Global solutions to a special case of the generalized weakly dissipative periodic two-component Camassa-Holm system, Nonlinear Anal., 117 (2015), pp. 38–46.CrossRefGoogle Scholar
[36]Rodrõguez-Blanco, G., On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), pp. 309–327.Google Scholar
[37]Olver, P. J. and Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53(2) (1996), pp. 1900–1906.CrossRefGoogle ScholarPubMed
[38]Qiao, Z., Yan, K., and Yin, Z., Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Comm. Math. Phys., 336(2) (2015), pp. 581–617.Google Scholar
[39]Rocca, G., Lombardo, M., Sammartino, M., and Sciacca, V., Singularity tracking for Camassa-Holm and Prandtl's equations, Appl. Numer. Math., 56 (2006), pp. 1108–1122.Google Scholar
[40]Xu, Y. and Shu, C.-W., A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal., 46 (2008), pp. 1998–2021.Google Scholar
[41]Tian, L., Xia, Z., and Zhang, P., Nonuniform continuity of the solution map to the two component Camassa-Holm system, J. Math. Anal. Appl. 416(1) (2014), pp. 374–389.CrossRefGoogle Scholar
[42]Yin, Z., On the Cauchy problem for an integrable equation with peakon solutions, Ill. J. Math., 47 (2003), pp. 649–666.Google Scholar