Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations (Q1889956)

The authors study the semilinear wave equation on the circle \(S^1=\mathbb{R}/2\pi \mathbb{Z}\): \[ \partial_{tt}u= \partial_{xx}u+f(u). \] In addition of possessing a Lagrangian and Hamiltonian structure, the evolution preserves the momentum \[ J[u](t)=\int_{S^1} \partial_t u\partial_x u\,dx. \] The authors find numerically that, among different spatial semi-discretization only the variational spatial semi-discretization on a uniform grid preserves an interpolated momentum within computational accuracy, while others schemes display a significant momentum drift. From the Contents: Variational framework, Variational semidiscretizations, Numerical results, Spatial discretization and momentum maps, Momentum error bounds.