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**Abstract**

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

Marcel Oliver (Bremen), Matthew West (CalTech), Claudia Wulff (Warwick, FU Berlin, Surrey)

**Abstract:** We prove that a standard second order finite difference uniform space
discretization of the nonlinear wave equation with periodic boundary
conditions, analytic nonlinearity, and analytic initial data conserves
moentum up to an error which is exponentially small in the
stepsize. Our estimates are valid for as long as the trajectories of
the full nonlinear wave equation remain real analytic.
The method of proof is that of backward error analysis, whereby we
construct a modified equation which is itself 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.
We also consider 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.

*This document has been created automatically by BIB 2.03*