Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-09T09:33:39.067Z Has data issue: false hasContentIssue false
  • English
  • Français

Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs

Part of: Numerical problems in dynamical systems Equations of mathematical physics and other areas of application Infinite-dimensional Hamiltonian systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

Wenjun Cai ,
Huai Zhang  and
Yushun Wang
Wenjun Cai
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Huai Zhang
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China
Yushun Wang*
Affiliation:
Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
*
*Corresponding author. Email addresses:wenjuncai1@gmail.com(W. Cai), hzhang@ucas.ac.cn(H. Zhang), wangyushun@njnu.edu.cn(Y. Wang)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

Keywords

Discrete singular convolutiondifferential matrixsymplectic integratorHamiltonian PDEs

MSC classification

Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 65P10: Hamiltonian systems including symplectic integrators 65M20: Method of lines 65M70: Spectral, collocation and related methods

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1375 - 1396
Copyright
Copyright © Global-Science Press 2016 

References

[1]Feng, K. and Qin, M.Z.. The symplectic methods for the computation of Hamiltonian equations. In Zhu, You-Ian and Guo, Ben-yu, editors, Numerical Methods for Partial Differential Equations, volume 1297 of Lecture Notes in Mathematics, pages 1-37. Springer Berlin Heidelberg, 1987.Google Scholar
[2]Sanz-Serna, J.M. and Calvo, M.P.. Numerical Hamiltonian problems. Chapman & Hall, 1994.Google Scholar
[3]Hairer, E., Lubich, C., and Wanner, G.. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin, 2002.Google Scholar
[4]McLachlan, R.I. and Quispel, G.R.W.. Geometric integrators for ODEs. J. Phys. A: Math. Gen., 39:5251,2006.CrossRefGoogle Scholar
[5]Ascher, U.M. and Mclachlan, R.I.. On symplectic and multisymplectic schemes for the KdV equation. J. Sci. Comput., 25:83–104,2005.CrossRefGoogle Scholar
[6]Wang, Y.S., Wang, B., Ji, Z.Z., and Qin, M.Z.. High order symplectic schemes for the sine-Gordon equation. J. Phys. Soc. Jpn., 72:2731–2736,2003.Google Scholar
[7]Wang, D.L.. Semi-discrete Fourier spectral approximations of infinite dimensional Hamiltonian system and conservation laws. Comput. Math. Appl., 21:63–75,1991.Google Scholar
[8]Chen, J.B.. Symplectic and multisymplectic Fourier pseudospectral discretizations for the Klein-Gordon equation. Lett. Math. Phys., 75:293–305,2006.Google Scholar
[9]Ma, J.W. and Yang, H.Z.. Multiresolution symplectic scheme for wave propagation in complex media. Appl. Math. Mech., 25:573–579,2004.Google Scholar
[10]Ma, J.W.. An exploration of multiresolution symplectic scheme for wave propagation using second-generation wavelets. Phys. Lett. A, 328:36–46,2004.Google Scholar
[11]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., 229:2550–2572,2010.CrossRefGoogle Scholar
[12]McLachlan, R.I.. Symplectic integration of Hamiltonian wave equations. Numer. Math., 66:465–492,1994.CrossRefGoogle Scholar
[13]Islas, A.L., Karpeev, D.A., and Schober, C.M.. Geometric integrators for the nonlinear Schrödinger equation. J. Comput. Phys., 173:116–148,2001.CrossRefGoogle Scholar
[14]Kong, L.H., Hong, J.L., Wang, L., and Fu, F.F.. Symplectic integrator for nonlinear high order Schrödinger equation with a trapped term. J. Comput. Appl. Math., 231:664–679,2009.Google Scholar
[15]Sun, Y.J. and Tse, P.S.P.. Symplectic and multisymplectic numerical methods for Maxwell's equations. J. Comput. Phys., 230:2076–2094,2011.CrossRefGoogle Scholar
[16]Sha, W., Huang, Z.X., Chen, M.S., and Wu, X.L.. Survey on symplectic finite-difference time-domain schemes for Maxwell's equations. IEEE Trans. Antennas Propag., 56:493–500,2008.Google Scholar
[17]Kong, L.H., Hong, J.L., Fu, F.F., and Chen, J.. Symplectic structure-preserving integrators for the two-dimensional Gross-Pitaevskii equation for BEC. J. Comput. Appl. Math., 235:4937–4948, 2011.Google Scholar
[18]Tian, Y.M. and Qin, M.Z.. Explicit symplectic schemes for investigating the evolution of vortices in a rotating Bose-Einstein Condensate. Comput. Phys. Commun., 155:132–143,2003.CrossRefGoogle Scholar
[19]Wei, G.W.. Discrete singular convolution for the solution of the Fokker-Planck equation. J. Chem. Phys., 110:8930–8942,1999.Google Scholar
[20]Wei, G.W.. A unified approach for solving the Fokker-Planck equations. J. Phys. A, 33:4935–4953, 2000.CrossRefGoogle Scholar
[21]Zhao, S. and Wei, G.W.. Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher's equation. SIAMJ. Sci. Comput., 25:127–147,2003.Google Scholar
[22]Yang, S.Y., Zhou, Y.C., and Wei, G.W.. Comparison of the discrete singular convolution algorithm and Fourier pseudospectral method for solving partial differential equations. Comput. Phys. Commun., 143:113–135,2002.CrossRefGoogle Scholar
[23]Wei, G.W.. Discrete singular convolution method for the sine-Gordon equation. Physica D, 137:247–259,2000.Google Scholar
[24]Feng, B.F. and Wei, G.W.. A comparison of the spectral and the discrete singular convolution schemes for the KdV type equations. J. Comput. Appl. Math., 145:183–188,2002.CrossRefGoogle Scholar
[25]Li, X.F., Wang, W.S., Lu, M.W., Zhang, M.G., and Li, Y.Q.. Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method. Geophys. J. Int., 188:1382–1392,2012.Google Scholar
[26]Zhao, S. and Wei, G.W.. A unified discontinuous Galerkin framework for time integration. Math. Method Appl. Sci., 37:1042–1071,2014.Google Scholar
[27]Chen, J.B. and Qin, M.Z.. Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation. Electron. Trans. Numer. Anal., 12:193–204,2001.Google Scholar
[28]Daubechies, I.. Ten Lectures on Wavelets. SIAM, 1992.CrossRefGoogle Scholar
[29]Xu, Y. and Shu, C.-W.. Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys., 205:72–97,2005.CrossRefGoogle Scholar
[30]Miles, J.W.. An envelope soliton problem. SIAMJ. Appl. Math., 41:227–230,1981.Google Scholar
[31]Herbst, B.M., Varadi, F., and Ablowitz, M.J.. Symplectic methods for the nonlinear Schrödinger equation. Math. Comput. Simul., 37:353–369,1994.Google Scholar
[32]Schober, C.M.. Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation. Phys. Lett. A, 259:140–151,1999.Google Scholar
[33]Ablowitz, M.J. and Segur, H.. Solitions and the Inverse Scattering Transform. SIAM, Philadelphia, 1981.CrossRefGoogle Scholar
[34]McLachlan, R.I.. Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics. Phys. Rev. E, 59:2393–2405,1999.Google Scholar
[35]Sun, J.Q. and Qin, M.Z.. Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system. Comput. Phys. Commun., 155:221–235,2003.Google Scholar
[36]Wang, Y.S. and Li, S.T.. New schemes for the coupled nonlinear Schrödinger equation. Int. J. Comput. Math., 87:775–787,2008.CrossRefGoogle Scholar
[37]Bridges, T.J. and Reich, S.. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. J. Comput. Phys., 157:473–499,2000.Google Scholar
[38]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., 279:80–102,2014.Google Scholar