Conservative finite difference schemes for the Degasperis-Procesi equation. (English) Zbl 1250.76138
The Degasperis-Procesi equation is a completely integrable shallow water equation. For this equation linear as well as nonlinear finite difference schemes are proposed. These schemes conserve at the same time two invariants associated with the bi-Hamiltonian of the equation. Furthermore, unique solvability of the schemes is proven, and numerical examples are given.
Reviewer: Thomas Sonar (Braunschweig)
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
discrete variational derivative method; shallow water equation; bi-Hamiltonian; unique solvabilityReferences:
| [1] | Miyatake, Y.; Matsuo, T., Conservative finite difference schemes for the Degasperis-Procesi equation, Trans. Japan Soc. Ind. Appl. Math., 20, 219-239 (2010), (in Japanese) |
| [2] | Degasperis, A.; Procesi, M., Asymptotic integrability, (Symmetry and Perturbation Theory (1999), World Scientific Publishing: World Scientific Publishing River Edge), 23-37 · Zbl 0963.35167 |
| [3] | Fuchssteiner, B.; Fokas, A. S., Symplectic structures, their bäcklund transformations and hereditary symmetries, Physica D, 4, 47-66 (1981) · Zbl 1194.37114 |
| [4] | Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521 |
| [5] | Camassa, R.; Holm, D. D.; Hyman, J., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1993) · Zbl 0808.76011 |
| [6] | Johnson, R. S., Camassa-Holm, Korteweg-de Vries and related models for water, J. Fluid Mech., 455, 63-82 (2002) · Zbl 1037.76006 |
| [7] | Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010 |
| [8] | Degasperis, A.; Holm, D. D.; Hone, A. N.W., A new integrable equation with peakon solutions, Theoret. and Math. Phys., 133, 1463-1474 (2002) |
| [9] | Coclite, G. M.; Karlsen, K. H., On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233, 60-91 (2006) · Zbl 1090.35142 |
| [10] | Coclite, G. M.; Karlsen, K. H., On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation, J. Differential Equations, 234, 142-160 (2007) · Zbl 1133.35028 |
| [11] | Lundmark, H., Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17, 169-198 (2007) · Zbl 1185.35194 |
| [12] | Yin, Z., Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283, 129-139 (2003) · Zbl 1033.35121 |
| [13] | Yin, Z., Global weak solution for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212, 182-194 (2004) · Zbl 1059.35149 |
| [14] | Escher, J.; Liu, Y.; Yin, Z., Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241, 457-485 (2006) · Zbl 1126.35053 |
| [15] | Furihara, D.; Mori, D., General derivation of finite difference schemes by means of a discrete variation, Trans. Japan Soc. Ind. Appl. Math., 8, 317-340 (1998), (in Japanese) |
| [16] | E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D.R.J. O’Neale, B. Owren, G.R.W. Quispel, Preserving Energy Resp. Dissipation in Numerical PDEs, Using the Average Vector Field Method, NTNU Preprint Series: Numerics No.7/2009. http://www.math.ntnu.no/preprint/numerics/2009/N7-2009.pdf; E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D.R.J. O’Neale, B. Owren, G.R.W. Quispel, Preserving Energy Resp. Dissipation in Numerical PDEs, Using the Average Vector Field Method, NTNU Preprint Series: Numerics No.7/2009. http://www.math.ntnu.no/preprint/numerics/2009/N7-2009.pdf |
| [17] | McLachlan, R. I.; Quispel, G. R.W.; Robidoux, N., Geometric integration using discrete gradients, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 357, 1021-1045 (1999) · Zbl 0933.65143 |
| [18] | Matsuo, T., Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations, J. Comput. Appl. Math., 218, 506-521 (2008) · Zbl 1147.65078 |
| [19] | Abdelgadir, A. A.; Yao, Y.; Fu, Y.; Huang, P., (A Difference Scheme for the Camassa-Holm Equation. A Difference Scheme for the Camassa-Holm Equation, Lecture Notes in Comput. Sci., vol. 4682 (2007)), 1287-1295 |
| [20] | K. Takeya, Conservative finite difference schemes for the Camassa-Holm equation, Master’s Thesis, Osaka University, 2007 (in Japanese).; K. Takeya, Conservative finite difference schemes for the Camassa-Holm equation, Master’s Thesis, Osaka University, 2007 (in Japanese). |
| [21] | K. Takeya, D. Furihata, Conservative Finite Difference Schemes for the Camassa-Holm Equation (in preparation).; K. Takeya, D. Furihata, Conservative Finite Difference Schemes for the Camassa-Holm Equation (in preparation). |
| [22] | Matsuo, T., A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation, J. Comput. Appl. Math., 234, 1258-1266 (2010) · Zbl 1203.65185 |
| [23] | Matsuo, T.; Yamaguchi, H., An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations, J. Comput. Phys., 228, 4346-4358 (2009) · Zbl 1169.65097 |
| [24] | Matsuo, T.; Furihata, D., Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171, 425-447 (2001) · Zbl 0993.65098 |
| [25] | Coclite, G. M.; Karlsen, K. H.; Risebro, N. H., Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28, 80-105 (2008) · Zbl 1246.76114 |
| [26] | Feng, B. F.; Liu, Y., An operator splitting method for the Degasperis-Procesi equation, J. Comput. Phys., 228, 7805-7820 (2009) · Zbl 1175.65094 |
| [27] | Hoel, H. A., A numerical scheme using multi-shock peakons to compute solutions of the Degasperis-Procesi equation, Electron. J. Differential Equations, 2007, 1-22 (2007) · Zbl 1133.35430 |
| [28] | Brezis, H., Analyse Fonctionnelle (1983), Masson: Masson Paris · Zbl 0511.46001 |
| [29] | Furihata, D., Finite difference schemes for \(\frac{\partial u}{\partial t} = (\frac{\partial}{\partial x})^\alpha \frac{\delta G}{\delta u}\) that inherit energy conservation or dissipation property, J. Comput. Phys., 156, 181-205 (1999) · Zbl 0945.65103 |
| [30] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0865.65009 |
| [31] | Constantin, A.; Molinet, L., Global weak solutions for a shallow water equation, Comm. Math. Phys., 211, 45-61 (2000) · Zbl 1002.35101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
