Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T15:02:19.802Z Has data issue: false hasContentIssue false
  • English
  • Français

An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs

Part of: Numerical methods in Fourier analysis Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  02 November 2016

Yuezheng Gong  and
Yushun Wang
Yuezheng Gong*
Affiliation:
Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
Yushun Wang*
Affiliation:
Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
*
*Corresponding author. Email addresses:[email protected] (Y. Gong), [email protected] (Y.Wang)
*Corresponding author. Email addresses:[email protected] (Y. Gong), [email protected] (Y.Wang)
Get access

Abstract

In this paper, we develop a novel energy-preserving wavelet collocation method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, the wavelet collocation method is conducted for spatial discretization. The obtained semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which has an energy conservation law. Then, the average vector field method is used for time integration, which leads to an energy-preserving method for multi-symplectic Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger equation and the Camassa-Holm equation. Since differentiation matrix obtained by the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform to solve equations in numerical calculation. Numerical experiments show the high accuracy, effectiveness and conservation properties of the proposed method.

Keywords

Energy-preservingaverage vector field methodwavelet collocation methodnonlinear Schrödinger equationCamassa-Holm equation

MSC classification

Secondary: 65M06: Finite difference methods 65M70: Spectral, collocation and related methods 65T50: Discrete and fast Fourier transforms 65Z05: Applications to physics
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1313 - 1339
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Diffenential Equations, Berlin: Springer-Verlag, 2002.CrossRefGoogle Scholar
[2] Feng, K. and Qin, M. Z., Symplectic Geometric Algorithms for Hamiltonian Systems, Berlin/Hangzhou: Springer-Verlag/Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, 2003.Google Scholar
[3] Bridges, T. J., Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147–190.CrossRefGoogle Scholar
[4] Marsden, J., Patrick, G. and Shkoller, S., Multi-symplectic Geometry, Variational Integrators and Nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351–395.CrossRefGoogle Scholar
[5] Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), 473–499.CrossRefGoogle Scholar
[6] Reich, S., Backward error analysis for numerical integrators, SIAM J.Numer. Anal., 36 (1999), 475–491.CrossRefGoogle Scholar
[7] Frank, J., Moore, B. E. and Reich, S., Linear PDEs and numerical methods that preserve a multi-symplectic conservation law, SIAM J. Sci. Comput., 28 (2006), 260–277.CrossRefGoogle Scholar
[8] Ryland, B. N. and Mclachlan, R. I., On multi-symplecticity of partitioned Runge-Kutta methods, SIAM J. Sci. Comput., 30 (2008), 1318–1340.Google Scholar
[9] Gong, Y. Z., Cai, J. X. and Wang, Y. S., Multi-symplectic Fourier pseudospectral method for the Kawahara equation, Commun. Comput. Phys., 16 (2014), 35–55.Google Scholar
[10] Bridges, T. J. and Reich, S., Numerical methods for Hamiltonian PDEs, J. Phys. A: Mathematical and General, 39 (2006), 5287–5320.Google Scholar
[11] Wang, Y. S. and Hong, J. L., Multi-symplectic algorithms for Hamiltonian partial differential equations, Comm. Appl. Math. Comput., 27 (2013), 163–230.Google Scholar
[12] Feynman, R. P., Conservation of Energy, The Feynman Lectures on Physics, vol. 1, Addison-Wesley Pub. Co., 1965, chapter 4.Google Scholar
[13] Benjamin, T. B., The stability of solitary waves, Proc. Roy. Soc. London A, 23 (1972), 153–183.Google Scholar
[14] Li, S. and Vu-Quoc, L., Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32 (1995), 1839–1875.Google Scholar
[15] Guo, B. Y. and Vázquez, L., A numerical scheme for nonlinear Klein-Gordon Equation, J. Appl. Sci., 1 (1983), 25–32.Google Scholar
[16] Fei, Z. and Vázquez, L., Two energy conserving numerical schemes for the Sine-Gordon equation, Appl. Math. Comput., 45 (1991), 17–30.Google Scholar
[17] Ringler, T. D., Thuburn, J., Klemp, J. B. and Skamarock, W. C., A unified approach to energy conservation and potential vorticity dynamics for arbitrarilystructured C-grids, J. Comput. Phys., 229 (2010), 3065–3090.CrossRefGoogle Scholar
[18] Chen, Y., Sun, Y. J. and Tang, Y. F., Energy-preserving numerical methods for Landau-Lifshitz equation, J. Phys. A: Math. Theor., 44 (2011), 295207.CrossRefGoogle Scholar
[19] Cai, J. X., Wang, Y. S. and Liang, H., Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schröinger system, J. Comput. Phys., 239 (2013), 30–50.Google Scholar
[20] Saito, N. and Beylkin, G., Multiresolution repersentations using the autocorrelation functions of compactly supported wavelets, IEEE Trans. Signal Process., 41 (1993), 3584–3590.CrossRefGoogle Scholar
[21] Vasilyev, O. V., Paolucci, S. and Sen, M., A multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. Comput. Phys., 120 (1995), 33–47.CrossRefGoogle Scholar
[22] Vasilyev, O. V. and Paolucci, S., A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comput. Phys., 138 (1997), 16–56.CrossRefGoogle Scholar
[23] Ma, J. W. and Yang, H. Z., Multiresolution symplectic scheme for wave propagation in complex media, Appl. Math. Mech., 25 (2004), 573–579.Google Scholar
[24] Ma, J. W., An exploration of multiresolution symplectic scheme for wave propagation using second-generation wavelets, Phys. Lett. A, 328 (2004), 36–46.CrossRefGoogle Scholar
[25] Bertoluzza, S. and Naldi, G., A wavelet collocation method for the numerical solution of partial differential equations, Appl. Comput. Harmon. Anal., 3 (1996), 1–9.CrossRefGoogle Scholar
[26] Zhu, H. J., Tang, L. Y., Song, S. H., Tang, Y. F. and Wang, D. S., Symplectic wavelet collocation method for Hamiltonian wave equations, J. Comput. Phys., 229 (2010), 2550–2572.CrossRefGoogle Scholar
[27] Zhu, H. J., Song, S. H. and Tang, Y. F., Multi-symplectic wavelet collocation method for the Schrödinger equation and the Camassa-Holm equation, Computer Physics Communications, 182 (2011), 616–627.CrossRefGoogle Scholar
[28] Mclachlan, R. I., Quispel, G. R. W. and Robidoux, N., Geometric integration using discrete gradients, Philos. Trans. R. Soc. A, 357 (1999), 1021–1046.CrossRefGoogle Scholar
[29] Quispel, G. R. W. and McLaren, D. I., A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206.Google Scholar
[30] Celledoni, E., Mclachlan, R. I., McLaren, D. I., Owren, B., Quispel, G. R. W. and Wright, W. M., Energy-preserving Runge-Kutta methods, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 645–649.CrossRefGoogle Scholar
[31] Hairer, E., Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math. 5 (2010), 73–84.Google Scholar
[32] Celledoni, E., Grimm, V., Mclachlan, R. I., McLaren, D. I., O’Neale, D., Owren, B. and Quispel, G. R. W., Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, J. Comput. Phys., 231 (2012), 6770–6789.CrossRefGoogle Scholar
[33] Gong, Y. Z., Cai, J. X. and Wang, Y. S., Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), 80–102.CrossRefGoogle Scholar
[34] Gray, R. M., Toeplitz and Circulant Matrices: A Review, 2006.CrossRefGoogle Scholar
[35] Moore, B. and Reich, S., Multisymplectic integration methods for Hamiltonian PDEs, J. Future Gener. Comput. Syst., 19 (2003), 395–402.CrossRefGoogle Scholar
[36] Wang, J., A note on multisymplectic Fourier pseudospectral discretization for the nonlinear Schrödinger equation, Appl. Math. Comput., 191 (2007), 31–41.Google Scholar
[37] Camassa, R. and Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664.Google Scholar
[38] Kalisch, H. and Lenells, J., Numerical study of traveling-wave solutions for the Camassa-Holm equation, Chaos Solitons Fractals, 25 (2005), 287–298.CrossRefGoogle Scholar
[39] Coclite, G. M., Karlsen, K. H. and Risebro, N. H., A Convergent Finite Difference Scheme for the Camassa-Holm Equation with General H 1 Initial Data, SIAM J. Numer. Anal., 46 (2008), 1554–1579.CrossRefGoogle Scholar
[40] Holden, H. and Raynaud, X., Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44 (2006), 1655–1680.Google Scholar
[41] Kalisch, H. and Raynaud, X., Convergence of a spectral projection of the Camassa-Holm equation, Numerical Methods for Partial Differential Equations, 22 (2006), 1197–1215.CrossRefGoogle Scholar
[42] Xu, Y. and Shu, C. W., A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal., 46 (2008), 1998–2021.CrossRefGoogle Scholar
[43] Xu, Y. and Shu, C.W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72–97.CrossRefGoogle Scholar
[44] Cohen, D., Owren, B. and Raynaud, X., Multi-symplectic integration of the Camassa-Holm equation, J. Comput. Phys., 227 (2008), 5492–5512.Google Scholar