I will discuss approximate momentum conservation for a second order finite difference uniform space discretization of the nonlinear wave equation with periodic boundary conditions.

The analysis employs backward error analysis. We construct a modified equation which, like the nonlinear wave equation, is Lagrangian and translation invariant, and therefore also conserves momentum. This modified equation interpolates the semidiscrete system for all time, and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system. These properties directly imply approximate momentum conservation for the semidiscrete system.

For comparison we also discuss discretizations that are not variational as well as discretizations on non-uniform grids. Through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum.