exploit the ellipticity of the Einstein-DeTurck equation and solve it via a diffusive flow. The canonical flow is Ricci-DeTurck flow (diffeomorphic to Ricci-flow), and the idea is to provide an initial guess and then simulate the flow, which hopefully asymptotes to the desired Ricci-flat fixed point. However for vacuum black hole spacetimes the Ricci-flat solution may not be an attractive fixed point of Ricci-flow, as such black holes usually possess Euclidean negative modes [@Headrick:2006ti]—in vacuum we expect that any black hole with a negative specific heat capacity is likely to have one or more negative modes [@Whiting:1988qr; @Whiting:1988ge; @Prestidge:1999uq; @Reall:2001ag; @Gregory:2001bd]. Other interesting Euclidean geometries also possess negative modes, and show analogous behaviour to black holes under Ricci-flow [@Holzegel:2007zz]. We therefore present an algorithm in which a finite-dimensional family of initial guess geometries must be chosen, and the parameters of this family varied in order to reach the fixed point. Since the Ricci-DeTurck flow is diffeomorphic to Ricci-flow, which provides a flow not only in the space of metrics but also in the space of diffeomorphism classes (or ‘geometries’), this method has the attractive feature that the algorithm is independent of the choice of reference connection. We denote this approach 