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Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. (English) Zbl 0984.37104
Summary: The symplectic numerical integration of finite-dimensional Hamiltonian systems is a well established subject and has led to a deeper understanding of existing methods as well as to the development of new very efficient and accurate schemes, e.g., for rigid body, constrained, and molecular dynamics. The numerical integration of infinite-dimensional Hamiltonian systems or Hamiltonian PDEs is much less explored. In this Letter, we suggest a new theoretical framework for generalizing symplectic numerical integrators for ODEs to Hamiltonian PDEs in $\bbfR^2$: time plus one space dimension. The central idea is that symplecticity for Hamiltonian PDEs is directional: the symplectic structure of the PDE is decomposed into distinct components representing space and time independently. In this setting PDE integrators can be constructed by concatenating uni-directional ODE symplectic integrators. This suggests a natural definition of multi-symplectic integrator as a discretization that conserves a discrete version of the conservation of symplecticity for Hamiltonian PDEs. We show that this approach leads to a general framework for geometric numerical schemes for Hamiltonian PDEs, which have remarkable energy and momentum conservation properties. Generalizations, including development of higher-order methods, application to the Euler equations in fluid mechanics, application to perturbed systems, and extension to more than one space dimension are also discussed.

37M15Symplectic integrators (dynamical systems)
37J05Relations of dynamical systems with symplectic geometry and topology
Full Text: DOI
[1] Abbott, M. B.: Computational hydraulics. (1979) · Zbl 0406.76002
[2] Abbott, M. B.; Basco, D. R.: Computational fluid dynamics. (1989) · Zbl 0743.76001
[3] Benettin, G.; Giorgilli, A.: J. stat. Phys.. 74, 1117 (1994)
[4] Binz, E.; Śniatycki, J.; Fischer, H.: Geometry of classical fields. (1988) · Zbl 0675.53065
[5] Bridges, T. J.: Math. proc. Cambridge philos. Soc.. 121, 147 (1997)
[6] Bridges, T. J.: Proc. R. Soc. London A. 453, 1365 (1997)
[7] Bridges, T. J.: Eur. J. Mech. B/fluids. 18, 493 (1999)
[8] Bridges, T. J.; Derks, G.: Proc. R. Soc. London A. 455, 2427 (1999)
[9] T.J. Bridges, S. Reich, in preparation (2001)
[10] Deuflhard, P.; Hermans, J.; Leimkuhler, B.; Mark, A. E.; Reich, S.; Skeel, R. D.: Lecture notes in computational science and engineering. 4 (1999) · Zbl 0904.00046
[11] García, P. L.: Sympos. math.. 14, 219 (1974)
[12] Gotay, M. J.: M.francaviglia mechanics, analysis and geometry: 200 years after Lagrange. Mechanics, analysis and geometry: 200 years after Lagrange, 203-235 (1991)
[13] Hairer, E.; Lubich, Ch.: Numer. math.. 76, 441 (1997)
[14] Kijowski, J.; Tulczyjew, W.: A symplectic framework for field theories. (1979) · Zbl 0439.58002
[15] Marsden, J. E.; Patrick, G. P.; Shkoller, S.: Comm. math. Phys.. 199, 351 (1999)
[16] Marsden, J. E.; Shkoller, S.: Math. proc. Cambridge philos. Soc.. 125, 553 (1999)
[17] Mclachlan, R. I.: Numer. math.. 66, 465 (1994)
[18] B. Moore, Ph.D. progress report, University of Surrey (2000)
[19] Reich, S.: SIAM numer. Anal.. 36, 1549 (1999)
[20] Reich, S.: J. comput. Phys.. 157, 473 (2000)
[21] Reich, S.: Bit. 40, 559 (2000)
[22] Sanz-Serna, J. M.; Calvo, M. P.: Numerical Hamiltonian systems. (1994) · Zbl 0816.65042