We present a new numerical dissipation algorithm, which can be efficiently used in combination with centered finite-difference methods. We start from a formulation of centered finite-volume methods for Numerical Relativity, in which third-order space accuracy can be obtained by employing just piecewise-linear reconstruction. We obtain a simplified version of the algorithm, which can be viewed as a centered finite-difference method plus some “adaptive dissipation”. The performance of this algorithm is confirmed by numerical results obtained from 3D black hole simulations.