**Approximate Momentum Conservation for Spatial
Semidiscretizations of Nonlinear Wave Equations**

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.

*Claudia Wulff*