Geometric Integrators for Nonlinear Hyperbolic Equations.

Abstract: In this paper an investigation of the Keller--Preissman box scheme applied to the sine--Gordon, NLS and cNLS equations is presented. The comparison between this scheme and other popular second--order integrators: Crank--Nicolson (symplectic in time) and leap--frog schemes is reported. The preservation of the Hamiltonian, momentum and the dispersion relation is analyzed for the discretizations of the sine--Gordon equation. A remarkably good behavior of both symplectic and multisymplectic schemes is found in the mcase of periodic boundary conditions. Some difficulties in correctly resolving the soliton solution for an infinite line case were observed for the box scheme. One question addressed in this work is how well the (linear) dispersion relation is preserved by the multisymplectic box scheme and what is the influence of the error in the numerical dispersion relation on the accuracy of the numerical solution.