Centered finite volume methods are considered in the context of numerical relativity. A specific formulation is presented, in which third-order space accuracy is reached by using a piecewise-linear reconstruction. This formulation can be interpreted as an “adaptive viscosity” modification of centered finite difference algorithms. These points are fully confirmed by one-dimensional black hole simulations. In the three-dimensional case, evidence is found that the use of a conformal decomposition is a key ingredient for the robustness of black hole numerical codes.