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

Authors:

Frédéric GibouAuthors Info & Claims

Journal of Computational Physics, Volume 228, Issue 16

Pages 5803 - 5818

https://doi.org/10.1016/j.jcp.2009.04.044

Published: 01 September 2009 Publication History

Abstract

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 L^1 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.

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Cited By

Larios-Cárdenas LGibou F(2023)Machine learning algorithms for three-dimensional mean-curvature computation in the level-set methodJournal of Computational Physics10.1016/j.jcp.2023.111995478:COnline publication date: 1-Apr-2023
https://dl.acm.org/doi/10.1016/j.jcp.2023.111995
Fullana TLe Chenadec VSayadi T(2023)Adjoint-based optimization of two-dimensional Stefan problemsJournal of Computational Physics10.1016/j.jcp.2022.111875475:COnline publication date: 15-Feb-2023
https://dl.acm.org/doi/10.1016/j.jcp.2022.111875
Peng ZAppelö DLiu S(2023)Universal AMG Accelerated Embedded Boundary Method Without Small Cell StiffnessJournal of Scientific Computing10.1007/s10915-023-02353-997:2Online publication date: 28-Sep-2023
https://dl.acm.org/doi/10.1007/s10915-023-02353-9
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A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations

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Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 228, Issue 16

September, 2009

460 pages

ISSN:0021-9991

Issue’s Table of Contents

Copyright © © 2009.

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Academic Press Professional, Inc.

United States

Publication History

Published: 01 September 2009

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Cited By

Larios-Cárdenas LGibou F(2023)Machine learning algorithms for three-dimensional mean-curvature computation in the level-set methodJournal of Computational Physics10.1016/j.jcp.2023.111995478:COnline publication date: 1-Apr-2023
https://dl.acm.org/doi/10.1016/j.jcp.2023.111995
Fullana TLe Chenadec VSayadi T(2023)Adjoint-based optimization of two-dimensional Stefan problemsJournal of Computational Physics10.1016/j.jcp.2022.111875475:COnline publication date: 15-Feb-2023
https://dl.acm.org/doi/10.1016/j.jcp.2022.111875
Peng ZAppelö DLiu S(2023)Universal AMG Accelerated Embedded Boundary Method Without Small Cell StiffnessJournal of Scientific Computing10.1007/s10915-023-02353-997:2Online publication date: 28-Sep-2023
https://dl.acm.org/doi/10.1007/s10915-023-02353-9
Bayat EEgan RBochkov DSauret AGibou F(2022)A sharp numerical method for the simulation of Stefan problems with convective effectsJournal of Computational Physics10.1016/j.jcp.2022.111627471:COnline publication date: 15-Dec-2022
https://dl.acm.org/doi/10.1016/j.jcp.2022.111627
Larios-Cárdenas LGibou F(2022)Error-correcting neural networks for semi-Lagrangian advection in the level-set methodJournal of Computational Physics10.1016/j.jcp.2022.111623471:COnline publication date: 15-Dec-2022
https://dl.acm.org/doi/10.1016/j.jcp.2022.111623
Larios-Cárdenas LGibou F(2022)A hybrid inference system for improved curvature estimation in the level-set method using machine learningJournal of Computational Physics10.1016/j.jcp.2022.111291463:COnline publication date: 15-Aug-2022
https://dl.acm.org/doi/10.1016/j.jcp.2022.111291
Larios-Cárdenas LGibou F(2022)Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-set MethodJournal of Scientific Computing10.1007/s10915-022-01952-293:1Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s10915-022-01952-2
Khalloufi MValette RHachem E(2020)Adaptive Eulerian framework for boiling and evaporationJournal of Computational Physics10.1016/j.jcp.2019.109030401:COnline publication date: 15-Jan-2020
https://dl.acm.org/doi/10.1016/j.jcp.2019.109030
Arias VBochkov DGibou F(2018)Poisson equations in irregular domains with Robin boundary conditions — Solver with second-order accurate gradientsJournal of Computational Physics10.1016/j.jcp.2018.03.022365:C(1-6)Online publication date: 15-Jul-2018
https://dl.acm.org/doi/10.1016/j.jcp.2018.03.022
Mistani PGuittet ABochkov DSchneider JMargetis DRatsch CGibou F(2018)The island dynamics model on parallel quadtree gridsJournal of Computational Physics10.1016/j.jcp.2018.01.054361:C(150-166)Online publication date: 15-May-2018
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