Stability of the DuFort–Frankel Scheme on Unstructured Grids

The DuFort–Frankel scheme was introduced in the 1950s to solve parabolic equations, and has been widely used ever since due to its stability and explicit nature. However, for over seven decades, its application has been limited to Cartesian grids. In this work, we propose a generalization of the DuFort–Frankel scheme that could be applied to arbitrary unstructured grids. Specifically, we focus on Voronoi grids in both 2D and 3D, and use the finite volume method for spatial discretization. Additionally, we present a proof of its stability based on the analysis of the spectrum of the amplification matrix, along with numerical examples.