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

Han Chen, Chohong Min, Frédéric Gibou

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51 Scopus citations


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.

Original languageEnglish
Pages (from-to)5803-5818
Number of pages16
JournalJournal of Computational Physics
Issue number16
StatePublished - 1 Sep 2009

Bibliographical note

Funding Information:
The research of H. Chen and F. Gibou was supported in part by a Sloan Research Fellowship in Mathematics, by the National Science Foundation Under Grant Agreement DMS 0713858 and by the Department of Energy Under Grant Agreement DE-FG02-08ER15991. The research of C. Min was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-331-C00045).


  • Level set
  • Non-graded adaptive grid
  • Quadtree
  • Stefan problem


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