geometric integrators, multisymplectic discretizations, discrete dispersion relationships, discrete wave action
This dissertation presents results of the study on symplectic and multisymplectic numerical methods for solving linear and nonlinear Hamiltonian wave equations. The emphasis is put on the second order space and time discretizations of the linear wave, the Klein-Gordon and the sine-Gordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and non-symplectic numerical methods. Tools used in the linear analysis are related to the Fourier transform and consist of the dispersion relationship and the power spectrum of the numerical solution. Nonlinear analysis, in turn, is closely connected to the temporal evolution of the total energy (Hamiltonian) and can be viewed from the topological perspective as preservation of the phase space structures. Using both linear and nonlinear diagnostics we find qualitative differences between MS and non-MS methods. The first difference can be noted in simulations of the linear wave equation solved for broad spectrum Gaussian initial data. Initial wave profiles of this type immediately split into an oscillatory wave-train with the high modes traveling faster (MS schemes), or slower (non-MS methods), than the analytic group velocity. This result is confirmed by an analysis of the dispersion relationship, which also indicates improved qualitative agreement of the dispersive curves for MS methods over non-MS ones. Moreover, observations of the convergence patterns in the wave profile obtained for the sine-Gordon equation for the initial data corresponding to the double-pole soliton and the temporal evolution of the Hamiltonian functional computed for solutions obtained from different discretizations suggest a change of the geometry of the phase space. Finally, we present some theoretical considerations concerning wave action. Lagrangian formulation of linear partial differential equations (PDEs) with slowly varying solutions is capable of linking the wave action conservation law with the dispersion relationship thus suggesting the possibility to extend this connection to multisymplectic PDEs.
Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Wlodarczyk, Tomasz, "Stability And Preservation Properties Of Multisymplectic Integrators" (2007). Electronic Theses and Dissertations. 3415.