×

Two simple numerical methods for the free boundary in one-phase Stefan problem. (English) Zbl 1442.80004

Summary: We present two simple numerical methods to find the free boundary in one-phase Stefan problem. The work is motivated by the necessity for better understanding of the solution surface (temperatures) near the free boundary. We formulate a log-transform function with the unfixed and fixed free boundary that has Lipschitz character near free boundary. We solve the quadratic equation in order to locate the free boundary in a time-recursive way. We also present several numerical results which illustrate a comparison to other methods.

MSC:

80A22 Stefan problems, phase changes, etc.
35R35 Free boundary problems for PDEs
80M50 Optimization problems in thermodynamics and heat transfer
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Stefan, J., Uber die theorie der eisbildung inbesondee uber die eisbindung im polarmeere, Annalen der Physik und Chemie, 42, 269-286 (1891) · JFM 23.1188.04
[2] Blanchard, D.; Porretta, A., Stefan problems with nonlinear diffusion and convection, Journal of Differential Equations, 210, 2, 383-428 (2005) · Zbl 1075.35112 · doi:10.1016/j.jde.2004.06.012
[3] Cheng, T.-F., Numerical analysis of nonlinear multiphase Stefan problems, Computers and Structures, 75, 2, 225-233 (2000) · doi:10.1016/S0045-7949(99)00071-1
[4] Juric, D.; Tryggvason, G., A front-tracking method for dendritic solidification, Journal of Computational Physics, 123, 1, 127-148 (1996) · Zbl 0843.65093 · doi:10.1006/jcph.1996.0011
[5] Hinze, M.; Ziegenbalg, S., Optimal control of the free boundary in a two-phase Stefan problem, Journal of Computational Physics, 223, 2, 657-684 (2007) · Zbl 1115.80008 · doi:10.1016/j.jcp.2006.09.030
[6] Liu, J.; Xu, M., Some exact solutions to Stefan problems with fractional differential equations, Journal of Mathematical Analysis and Applications, 351, 2, 536-542 (2009) · Zbl 1163.35043 · doi:10.1016/j.jmaa.2008.10.042
[7] Voller, V. R.; Swenson, J. B.; Paola, C., An analytical solution for a Stefan problem with variable latent heat, International Journal of Heat and Mass Transfer, 47, 24, 5387-5390 (2004) · Zbl 1077.80004 · doi:10.1016/j.ijheatmasstransfer.2004.07.007
[8] Yigit, F., Approximate analytical and numerical solutions for a two-dimensional Stefan problem, Applied Mathematics and Computation, 202, 2, 857-869 (2008) · Zbl 1147.65068 · doi:10.1016/j.amc.2008.03.033
[9] Lorenzo-Trueba, J.; Voller, V. R., Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, Journal of Mathematical Analysis and Applications, 366, 2, 538-549 (2010) · Zbl 1183.86002 · doi:10.1016/j.jmaa.2010.01.008
[10] Asaithambi, A., Numerical solution of Stefan problems using automatic differentiation, Applied Mathematics and Computation, 189, 1, 943-948 (2007) · Zbl 1122.65090 · doi:10.1016/j.amc.2006.11.159
[11] Nedoma, J., Numerical solution of a Stefan-like problem in Bingham rheology, Mathematics and Computers in Simulation, 61, 3-6, 271-281 (2003) · Zbl 1205.76161 · doi:10.1016/S0378-4754(02)00083-6
[12] Słota, D., Using genetic algorithms for the determination of an heat transfer coefficient in three-phase inverse Stefan problem, International Communications in Heat and Mass Transfer, 35, 2, 149-156 (2008) · doi:10.1016/j.icheatmasstransfer.2007.08.010
[13] Landau, H. G., Heat conduction in a melting solid, Quarterly Journal of Applied Mathematics, 8, 81-94 (1950) · Zbl 0036.13902
[14] Kutluay, S.; Bahadir, A. R.; Özdeş, A., The numerical solution of one-phase classical Stefan problem, Journal of Computational and Applied Mathematics, 81, 1, 135-144 (1997) · Zbl 0885.65102
[15] Kim, B. J.; Ahn, C.; Choe, H. J., Direct computation for American put option and free boundary using finite difference method, Japan Journal of Industrial and Applied Mathematics, 30, 1, 21-37 (2013) · Zbl 1258.91219 · doi:10.1007/s13160-012-0094-9
[16] Kim, B. J.; Ma, Y.-K.; Choe, H. J., A simple numerical method for pricing an American put option, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.91104 · doi:10.1155/2013/128025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.