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**An accurate finite-difference method for ablation-type Stefan problems.**
*(English)*
Zbl 1247.65119

Summary: A recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is extended for the purpose of solving one-phase ablation-type moving boundary problems; in tandem with the Keller box finite-difference scheme, the so-called boundary immobilization method is used. An important component of the work is the use of variable transformations that must be built into the numerical algorithm in order to preserve second-order accuracy in both time and space. The analysis also determines that the ablation front initially moves as the time raised to the power 3/2; hence, it evolves considerably more slowly than the phase-change front in the classical Stefan problem with isothermal cooling.

### MSC:

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

65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |

35K05 | Heat equation |

35R37 | Moving boundary problems for PDEs |

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

80M20 | Finite difference methods applied to problems in thermodynamics and heat transfer |

### Keywords:

ablation; Stefan problem; Keller box scheme; boundary immobilization; starting solutions; heat equation; numerical examples; algorithm; moving boundary problem; finite-difference scheme; isothermal cooling
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\textit{S. L. Mitchell} and \textit{M. Vynnycky}, J. Comput. Appl. Math. 236, No. 17, 4181--4192 (2012; Zbl 1247.65119)

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### References:

[1] | Andrews, J. G.; Atthey, D. R., On the motion of an intensely heated evaporating boundary, J. Inst. Math. Appl., 15, 59-72 (1975) |

[2] | Chen, Y.-K.; Milos, F. S., Ablation and thermal response program for spacecraft heatshield analysis, J. Spacecraft Rockets, 36, 3, 475-483 (1999) |

[3] | Campbell, A. J.; Humayun, M., Trace element microanalysis in iron meteorites by laser ablation ICPMS, Anal. Chem., 71, 5, 939-946 (1999) |

[4] | Lin, W.-S., Steady ablation on the surface of a two-layer composite, Int. J. Heat Mass Transf., 48, 5504-5519 (2005) · Zbl 1188.76270 |

[5] | Tayler, A. B., Mathematical Models in Applied Mechanics (2001), Oxford University Press · Zbl 0994.35001 |

[6] | Johnson, W. D., The profile of maturity in apline glacial erosion, J. Geology, 12, 7, 569-578 (1904) |

[7] | Namenanee, K.; McKenzie, J.; Kosa, E.; Schwab, M.; Sunsaneewitayakul, B.; Vasavakul, T.; Khunnawat, C.; Ngarmukos, T., A new approach for catheter ablation of atrial fibrillation: mapping of the electrophysiologic substrate, J. Amer. College Cardiology, 43, 11, 2044-2053 (2004) |

[8] | Landau, H. G., Heat conduction in a melting solid, Quart. Appl. Math., 8, 81-94 (1950) · Zbl 0036.13902 |

[9] | Goodman, T. R., Application of integral methods to transient nonlinear heat transfer, Adv. Heat Transf., 1, 51-122 (1964) |

[10] | Zien, T.-F., Integral solutions of ablation problems with time-tependent heat flux, AIAA J., 16, 1287-1296 (1978) |

[14] | Mitchell, S. L.; Myers, T. G., A heat balance integral method for one-dimensional finite ablation, AIAA J. Thermophysics, 22, 3, 508-514 (2008) |

[15] | Mitchell, S. L.; Myers, T. G., Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Rev., 52, 57-86 (2010) · Zbl 1188.80004 |

[16] | Mitchell, S. L., Applying the combined integral method to one-dimensional ablation, Appl. Math. Modelling, 36, 127-138 (2012) · Zbl 1236.80006 |

[17] | Myers, T. G., Optimal exponent heat balance and refined integral methods applied to Stefan problems, Int. J. Heat Mass Transf., 53, 1119-1127 (2010) · Zbl 1183.80091 |

[18] | Yang, L.; Zhang, Y.; Chen, J. K., An integral approximate solution to ablation of a two-layer composite with a temporal Gaussian heat flux, Heat Transf. Engng., 32, 5, 418-428 (2011) |

[19] | Storti, M., Numerical modeling of ablation phenomena as two-phase Stefan problems, Int. J. Heat Mass Transf., 38, 15, 2843-2854 (1995) · Zbl 0923.76333 |

[20] | Wang, J.; Wang, H.; Sun, J.; Wang, J., Numerical simulation of control ablation by transpiration cooling, Heat Mass Transf., 43, 471-478 (2007) |

[21] | Wong, S. K.; Walton, A., Numerical solution of single-phase problem using a fictitious material, Num. Heat Transf. B, 35, 211-223 (1999) |

[22] | Blackwell, B. F., Numerical prediction of one-dimensional ablation using a finite control volume procedure with exponential differencing, Num. Heat Transf., 14, 17-34 (1988) · Zbl 0667.76123 |

[23] | Blackwell, B. F.; Hogan, R. E., One-dimensional ablation using Landau transformation and finite control volume procedure, J. Thermophys. & Heat Transf., 8, 2, 282-287 (1994) |

[24] | Mitchell, S. L.; Vynnycky, M., Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems, Appl. Math. Comp., 215, 1609-1621 (2009) · Zbl 1177.80078 |

[25] | Mitchell, S. L.; Vynnycky, M.; Gusev, I. G.; Sazhin, S. S., An accurate numerical solution for the transient heating of an evaporating droplet, Appl. Math. Comp., 217, 9219-9233 (2011) · Zbl 1223.80009 |

[26] | Huppert, H. E., Phase changes following the initiation of a hot turbulent flow over a cold solid surface, J. Fluid Mech., 198, 293-319 (1989) · Zbl 0662.76072 |

[27] | King, J. R.; Riley, D. S., Asymptotic solutions to the Stefan problem with a constant heat source at the moving boundary, Proc. R. Soc. Lond. Ser. A, 456, 1163-1174 (2000) · Zbl 0973.35202 |

[29] | Carslaw, H. S.; Jaeger, J. C., Conduction of Heat in Solids (1947), Oxford University Press: Oxford University Press London · Zbl 0029.37801 |

[30] | Åberg, J.; Vynnycky, M.; Fredriksson, H., Heat-flux measurements of industrial on-site continuous copper casting and their use as boundary conditions for numerical simulations, Trans. Ind. Inst. Met., 62, 443-446 (2009) |

[31] | Vynnycky, M., A mathematical model for air-gap formation in vertical continuous casting: the effect of superheat, Trans. Ind. Inst. Met., 62, 495-498 (2009) |

[32] | Vynnycky, M., Concerning closed-streamline flows with discontinuous boundary conditions, J. Engrg. Math., 33, 141-156 (1998) · Zbl 0902.76023 |

[33] | Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations (2004), Society for Industrial Mathematics · Zbl 1071.65118 |

[34] | Roday, A. P.; Kazmierczak, M. J., Analysis of phase-change in finite slabs subjected to convective boundary conditions: part I—melting, Int. Rev. Chem. Eng. (Rapid Communications), 1, 87-99 (2009) |

[35] | Roday, A. P.; Kazmierczak, M. J., Analysis of phase-change in finite slabs subjected to convective boundary conditions: part II—freezing, Int. Rev. Chem. Eng. (Rapid Communications), 1, 100-108 (2009) |

[36] | Roday, A. P.; Kazmierczak, M. J., Melting and freezing in a finite slab due to a linearly decreasing free-stream temperature of a convective boundary condition, Thermal Sci., 2, 141-153 (2009) |

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