A more accurate two-dimensional grain growth algorithm

We describe a method for evolving two-dimensional polycrystalline microstructures via mean curvature flow that satisfies the von Neumann-Mullins relation with an absolute error O (Δt²). This is a significant improvement over a different method currently used that has an absolute error O (Δt). We describe the implementation of this method and show that while both approaches lead to indistinguishable evolution when the spatial discretization is very fine, the differences can be substantial when the discretization is left unrefined. We demonstrate that this new front-tracking approach can be pushed to the limit in which the only mesh nodes are those coincident with triple junctions. This reduces the method to a vertex model that is consistent with the exact kinetic law for grain growth. We briefly discuss an extension of the method to higher spatial dimensions.