| [1] Marsden J E and Shkoller S. Multisymplectic geometry, covariant Hamiltonians, and water waves[J]. Proc. Comb. Phil. Soc., 1999, 125:553-575.[2] Bridges T J. Multi-symplectic structures and wave propagation[J]. Math. Proc. Camb. Phil. Soc., 1997, 121:147-190.[3] Bridges T J and Reich S. Multi-symplectic integrators:Numerical schemes for Hamiltonian PDEs that conserve symplecticity[J]. Phys. Lett. A., 2001, 284:184-193.[4] Reich S. Multi-symplectic Runge-Kutta methods for Hamiltonian wave equation[J]. Comput. Phys., 2000, 157:473-499.[5] Bridges T J and Reich S. Numerical methods for Hamiltonian PDEs[J]. Phys A:Math. Gen., 2006, 39(19):5287-5320.[6] Bridges T J and Reich S. Multi-symplectic spectral discretization for the Zakharov-Kaznetsov and shallow water equation[J]. Physica D., 2001, 152:491-504.[7] Islas A L and Schober C M. Multi-symplectic methods for generalized schrodinger equation[J]. Future Gener. Comput. Syst., 2003, 19:403-413.[8] Hong J L and Li C. Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations[J]. Comput. Phys., 2006, 211:448-472.[9] Wang Y S and Hong J L. Multi-symplectic algorithms for Hamiltonian partial differential equation[J]. Commun. Appl. Math. Comput., 2014, 27:163-230.[10] McLachlan R I, Quispel G R W and Robidoux N. Geometric integration using discrete gradients. Phil. Trans. Roy. Soc. A., 1999, 357:1021-1046.[11] Quispel G R W and McLaren D I. A new class of energy-preserving numerical integration methods[J]. Phys. A., 2008, 41:045206.[12] Matsuo T. New conservative schemes with discrete varivational derivatives for nonlinear wave equations[J]. Comput. Appl. Math., 2007, 203(1):32-56.[13] Furihata D. Finite difference schemes for ∂u/∂t=(∂/∂x)α∂G/∂u that inherit energy conservation or dissipation property[J]. Comput. Phys., 1999, 156(1):181-205.[14] Brugnano L, Iavernaro F and Trigiante D. Hamiltonian boundary value methods (Energy preserving discrete line integral methods)[J]. Numeri. Anal., Industrial and Appl. Math., 2010, 5(1-2):17-37.[15] Brugnano L, Iavernaro F and Trigiante D. A note on the efficient implementation of Hamiltonian BVMs[J]. Comput. and Appl. Math., 2011, 236:375-383.[16] Wang Y S, Cai J X. A conservative fourier pseudospectral algorithm for a coupled nonline schroedinger system[J]. Chinese Physics B., 2013, 22(6):135-140.[17] Gong Y Z, Cai J X and Wang Y S. Some new structure-preserving algorithms for general multisymplectic formulation of Hamitonian PDEs[J]. Computational Physics., 2014, 279:80-102.[18] Yang Y H, Wang Y S and Song Y Z. A new local energy-preserving algorithm for the BBM equation[J]. Appl. Math. and Comput., 2018, 324:119-130.[19] Zhang H, Song S H Chen X D and Zhou W E. Average vector field methods for the coupled Schrödinger-KdV equations[J]. Chin. Phys. B., 2014, 23(7):070208.[20] Jiang C L, Sun J Q, He X F and Zhou L L. High order energy-preserving method of the "Good" Boussinesq equation[J]. Numer. Math. Theor. Meth. Appl., 2016, 9(1):111-112.[21] Jiang C L, Sun J Q, Li H C and Wang Y F.A fourth-order AVF method for the numerical integration of sine-Gordon equation[J]. Appl. Math. Comput., 2017, 313:144-158.[22] Chen J B, Qin M Z. Multi-synplectic Foueier pseudospectral method for the Schrödinger equation[J]. Electr. Numer. Anal., 2001, 12:193-204.[23] Wang J. A note on multi-symplectic Fourier pseudospectral discretization for the nonlinear Schrödinger equation[J]. Appl. Math. Comput., 2007, 191:31-41.[24] Alvarez A. Linear Crank-Nicholsen scheme for nonlinear Dirac equations[J]. J. Comput. Phys., 1992, 99:348-350. |
