Models of physical phenomena such as crystal growth or water waves generally involve complex moving interfaces, with velocities determined by interfacial geometry and material physics. Numerical methods for such models tend to be customized. As a consequence, they must be redesigned whenever the model changes. We present a general computational algorithm for evolving complex interfaces which treats the velocity as a black box, thus avoiding model-dependent issues. The interface is implicitly updated via an explicit second-order semi-Lagrangian advection formula which converts moving interfaces to a contouring problem. Spatial and temporal resolutions are decoupled, permitting grid-free adaptive refinement of the interface geometry. A 2D modular implementation computes highly accurate solutions to geometric moving interface problems involving merging, anisotropy, faceting, curvature, dynamic topology

and nonlocal interactions. The implementation couples with a Laplace solver based on classical potential theory, to provide fast accurate solutions of the Ostwald ripening model for grain growth by volume diffusion