Tomasz Wlodarczyk, University of Central Florida

Multisymplectic Integrators for Hamiltonian Partial Differential Equations

Abstract: One of the crucial properties of every Hamiltonian dynamical system is its symplecticity. It is therefore natural to construct numerical integrators preserving this property. In this talk the notion of symplecticity of Hamiltonian systems and numerical schemes will be defined and the discussion of some properties of finite–difference geometric (symplectic and multisymplectic) integrators will be presented. The central point of this discussion is the presentation of the performance diagnostics of the multisymplectic box scheme discretization of the sine–Gordon equation and comparison to other, symplectic and nonsymplectic, integrators of second order. Presented are results of numerical simulations of some soliton solutions to the sine–Gordon equation and the effect of perturbations introduced by the numerical integrators. Effects due to perturbations are analyzed qualitatively and quantitatively through numerically calculated error in the energy and the momentum functionals as well as local and global conservation laws. The stability conditions for this scheme can be obtained from dissipation–dispersion analysis via the notion of the symbol of the numerical scheme. Important phenomenon of qualitative change of the wave profile depending upon the choice of numerical integrator found in the simulation results is related to the structural stability of integrators and will be a subject of further studies. This is joint work with A. Islas, C. M. Schober, T. H. Wlodarczyk.

Advisor: Constance Schober (UCF)