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**Explicit variable time-step method for implicit moving boundary problems.**
*(English)*
Zbl 0801.65122

A one-dimensional diffusion equation \(\partial u/ \partial t = \partial^ 2 u/ \partial x^ 2 - 1\) with appropriate initial and boundary conditions and with the conditions \(u = 0\) and \(\partial u/ \partial x = 0\) on the unknown moving boundary \(x = s(t)\) is considered, which is like a one-phase Stefan problem but with vanishing latent heat. Such problem models the diffusion of oxygen into an absorbing medium.

An unconditionally stable explicit easily implementable finite difference numerical scheme is presented and tested in the paper. A comparison with results obtained by other methods of other authors is made, the results being similar but calculated here much more effectively.

An unconditionally stable explicit easily implementable finite difference numerical scheme is presented and tested in the paper. A comparison with results obtained by other methods of other authors is made, the results being similar but calculated here much more effectively.

Reviewer: T.Roubíček (Praha)

### MSC:

65Z05 | Applications to the sciences |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

80A22 | Stefan problems, phase changes, etc. |

35R35 | Free boundary problems for PDEs |

35K05 | Heat equation |

### Keywords:

explicit variable time-step method; implicit moving boundary; explicit finite difference discretization; diffusion equation; one-phase Stefan problem
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\textit{M. Zerroukat} and \textit{C. R. Chatwin}, Commun. Numer. Methods Eng. 10, No. 3, 227--235 (1994; Zbl 0801.65122)

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### References:

[1] | Crank, A moving boundary problem arising from the diffusion of oxygen in absorbing tissue, J. Inst. Math. Appl. 10 pp 19– (1972) · Zbl 0247.65064 |

[2] | Crank, A method for solving moving boudnary problems in heat flow using cubic splines or polynomials, J. Int. Math. Appl. 10 pp 269– (1972) · Zbl 0299.65049 |

[3] | D. H. Ferris S. Hill On the numerical solution of a one-dimensional diffusion problem with a moving boundary 1974 |

[4] | Berger, Numerical solution of a diffusion consumption problem with free boundary, SIAM J. Numer. Anal. 12 pp 646– (1975) · Zbl 0317.65032 |

[5] | Miller, A finite element moving boundary computation with adaptive mesh, J. Inst. Math. Appl. 22 pp 467– (1978) · Zbl 0394.65032 |

[6] | Hansen, On a moving boundary problem from biomechanics, J. Inst. Math. Appl. 13 pp 385– (1974) · Zbl 0307.45016 |

[7] | Furzeland, A comparative study of numerical methods for moving boundary problems, J. Inst. Math. Appl. 26 pp 411– (1980) · Zbl 0468.65063 |

[8] | Free and Moving Boundary Problems (1984) · Zbl 0547.35001 |

[9] | Gupta, Complete numerical solution of the oxygen diffusion problem involving a moving boundary, Comput. Methods Appl. Mech. Eng. 29 pp 233– (1981) · Zbl 0469.65087 |

[10] | Zerroukat, An explicit variable time step method for one-dimensional Stephan problems, Int. j. numer. methods eng. 35 pp 1503– (1992) · Zbl 0788.73079 |

[11] | Gupta, A modified variable time step method for one-dimensional Stephan problems, Comput. Methods Appl. Mech. Eng. 4 pp 143– (1974) · Zbl 0284.76072 |

[12] | Dahmardah, A Fourier series solution of the Crank-Gupta equation, IMA J. Numer. Anal. 3 pp 81– (1983) · Zbl 0516.65090 |

[13] | Bhattacharya, An explicit conditionally stable finite difference equation for heat conduction problems, Int. J. Numer. Methods Eng. 21 pp 239– (1985) · Zbl 0555.65064 |

[14] | Bhattacharya, Finite difference solutions of partial differential equations, Commun. Appl. Numer. Methods 6 pp 173– (1990) · Zbl 0704.65062 |

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