Multisymplecticity of the centred box discretization for Hamiltonian PDEs with m ≥ 2 space dimensions

Applied Mathematics Letters

Volumn 15, Issue 8, 2002, Pages 1005-1011

  Hong, Jialin ^a   Qin, Meng Zhao ^a  

^a INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

Author keywords

Centred box scheme; Hamiltonian PDEs; Multisymplecticity

Indexed keywords

EID: 31244436460     PISSN: 08939659     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0893-9659(02)00077-0     Document Type: Article

References (21)

1. T.J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Camb. Phil. Soc. 121, 147-190, (1997).
2. T.J. Bridges, A geometrical formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proc. Roy. Soc. Lond. A 453, 1365-1395, (1997).
3. T.J. Bridges, A new framework for studying the stability of genus-1 and genus-2 KP patterns, Euro. J. Mech. B/Fluids 18, 493-500, (1999).
4. T.J. Bridges and G. Derks, Unstable eigenvalues, and the linearization about solitary waves and fronts with symmetry, Proc. Roy. Soc. Lond. A 455, 2427-2469, (1999).
5. T.J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, (preprint).
6. S. Reich, Multi-symplectic Runge-Kutta methods for Hamiltonian wave equations, J. Comput. Phy. 157, 473-499, (2000).
7. T.T. Liu and Y.S. Wang, Multi-symplectically numerical treatment of KP equation, (in preparation).
8. J.E. Marsden, G.P. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. in Math. Phys. 199, 351-395, (1999).
9. J.E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Comb. Phil. Soc. 125, 553-575, (1999).
10. P.F. Zhao and M.Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phy. A: Math. Gen. 33, 3613-3626, (2000).
11. M.J. Ablowitz and P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society, Lecture Notes Series, Volume 149, Cambridge University Press, Cambridge, MA, (1991).
12. A. deBouard and J.C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal. 28, 1064-1085, (1997).
13. X.L. Dou and G.Q. Zhang, On an inverse problem for 2-D wave equation, In Proceeding of Conf. Sci. & Eng. Comp. (for Young Chinese Scientists), National Defence Industry Press, Beijing, (1994).
14. P.G. Drazin and R.S. Johnson, Solitons: An Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, (1989).
15. B.F. Feng and T. Mitsui, A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations, J. Comp. & Appl. Math. 90, 95-116, (1998).
16. N.C. Freeman, Soliton interactions in two dimensions, Adv. Appl. Mech. 20, 1-37, (1980).
17. A.A. Minzoni and N.F. Smyth, Evolution of lump solutions for the KP equation, Wave Motion 24, 291-305, (1996).
18. H. Segur and A. Finkel, An analytical model of periodic waves in shallow water, Stud. Appl. Math. 73, 183-220, (1985).
19. S.B. Wineberg, J.F. McGrath, E.F. Gabi, L.R. Scott and E. Southwell, Implicit spectral methods for wave propagation problems, J. Comput. Phy. 97, 311-336, (1991).
20. X.P. Wang, M.J. Ablowitz and H. Segur, Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petvishvili equation, Physica D 78, 241-265, (1994).
21. B.B. Kadomtsev and V.I. Petviashvili, On the stability of solitary waves in a weakly dispersive media, Sov. Phys. Dokl. 15, 539-541, (1970).