  The scheduled launch of the LISA Mission in the next decade has called
attention to the gravitational self-force problem. Despite an extensive body of
theoretical work, long-time numerical computations of gravitational waves from
extreme-mass-ratio-inspirals remain challenging. This work proposes a class of
numerical evolution schemes suitable to this problem based on Hermite
integration. Their most important feature is time-reversal symmetry and
unconditional stability, which enables these methods to preserve symplectic
structure, energy, momentum and other Noether charges over long time periods.
We apply Noether's theorem to the master fields of black hole perturbation
theory on a hyperboloidal slice of Schwarzschild spacetime to show that there
exist constants of evolution that numerical simulations must preserve. We
demonstrate that time-symmetric integration schemes based on a 2-point Taylor
expansion (such as Hermite integration) numerically conserve these quantities,
unlike schemes based on a 1-point Taylor expansion (such as Runge-Kutta). This
makes time-symmetric schemes ideal for long-time EMRI simulations.
