A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate

We present a level set approach to the numerical simulation of the Stefan problem on non-graded adaptive Cartesian grids, i.e. grids for which the size ratio between adjacent cells is not constrained. We use the quadtree data structure to discretize the computational domain and a simple recursive algorithm to automatically generate the adaptive grids. We use the level set method on quadtree of Min and Gibou [C. Min, F. Gibou, A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys. 225 (2007) 300-321] to keep track of the moving front between the two phases, and the finite difference scheme of Chen et al. [H. Chen, C. Min, F. Gibou, A supra-convergent finite difference scheme for the poisson and heat equations on irregular domains and non-graded adaptive Cartesian grids, J. Sci. Comput. 31 (2007) 19-60] to solve the heat equations in each of the phases, with Dirichlet boundary conditions imposed on the interface. This scheme produces solutions that converge supralinearly (∼ 1.5) in both the L¹ and the L^∞ norms, which we demonstrate numerically for both the temperature field and the interface location. Numerical results also indicate that our method can simulate physical effects such as surface tension and crystalline anisotropy. We also present numerical data to quantify the saving in computational resources.