Multi-symplectic integration methods for Hamiltonian PDEs

Recent results on numerical integration methods that exactly preserve the symplectic structure in both space and time for Hamiltonian PDEs are discussed. The Preissman box scheme is considered as an example, and it is shown that the method exactly preserves a multi-symplectic conservation law and any conservation law related to linear symmetries of the PDE. Local energy and momentum are not, in general, conserved exactly, but semi-discrete versions of these conservation laws are. Then, using Taylor series expansions, one obtains a modified multi-symplectic PDE and modified conservation laws that are preserved to higher order. These results are applied to the nonlinear Schrodinger (NLS) equation and the sine-Gordon equation in relation to the numerical approximation of solitary wave solutions.

[1]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[2]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[3]  A. Durán,et al.  The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation , 2000 .

[4]  Ernst Hairer,et al.  The life-span of backward error analysis for numerical integrators , 1997 .

[5]  S. Reich.  Backward Error Analysis for Numerical Integrators , 1999 .

[6]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[7]  P. Drazin.  SOLITONS, NONLINEAR EVOLUTION EQUATIONS AND INVERSE SCATTERING (London Mathematical Society Lecture Note Series 149) , 1993 .

[8]  J. Sanz-Serna,et al.  Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation , 1997 .

[9]  G. Benettin,et al.  On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms , 1994 .

[10]  T. Bridges.  Multi-symplectic structures and wave propagation , 1997, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Thomas J. Bridges,et al.  Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  S. Reich.  Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations , 2000 .

[13]  T. Bridges,et al.  The Symplectic Evans Matrix,¶and the Instability of Solitary Waves and Fronts , 2001 .

[14]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[15]  Michael B. Abbott,et al.  Computational Hydraulics , 1998 .

[16]  Brian E. Moore.  A modified equations approach for multi-symplectic integration methods. , 2003 .

[17]  Benedict Leimkuhler,et al.  Computational Molecular Dynamics: Challenges, Methods, Ideas , 1999, Computational Molecular Dynamics.

[18]  T. Bridges.  A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.