zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems. (English) Zbl 1177.80078
Summary: Although the numerical solution of one-dimensional phase-change, or Stefan, problems is well documented, a review of the most recent literature indicates that there are still unresolved issues regarding the start-up of a computation for a region that initially has zero thickness, as well as how to determine the location of the moving boundary thereafter. This paper considers the so-called boundary immobilization method for four benchmark melting problems, in tandem with three finite-difference discretization schemes. We demonstrate a combined analytical and numerical approach that eliminates completely the ad hoc treatment of the starting solution that is often used, and is numerically second-order accurate in both time and space, a point that has been consistently overlooked for this type of moving-boundary problem.
MSC:
80A22Stefan problems, phase changes, etc.
80M20Finite difference methods (thermodynamics)
References:
[1]Caldwell, J.; Kwan, Y. Y.: Numerical methods for one-dimensional Stefan problems, Commun. numer. Meth. eng. 20, 535-545 (2004) · Zbl 1048.65095 · doi:10.1002/cnm.691
[2]Esen, A.; Kutluay, S.: A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method, App. math. Comput. 148, 321-329 (2004) · Zbl 1034.65070 · doi:10.1016/S0096-3003(02)00846-9
[3]Caldwell, J.; Chan, C. -C.: Spherical solidification by the enthalpy method and the heat balance integral method, Appl. math. Modell. 24, 45-53 (2000) · Zbl 0944.80001 · doi:10.1016/S0307-904X(99)00031-1
[4]Caldwell, J.; Savović, S.: Numerical solution of Stefan problem by variable space grid and boundary immobilization method, J. math. Sci. 13, 67-79 (2002)
[5]Caldwell, J.; Kwan, Y. Y.: Starting solutions for the boundary immobilization method, Commun. numer. Meth. eng. 21, 289-295 (2005) · Zbl 1112.80011 · doi:10.1002/cnm.747
[6]Savović, S.; Caldwell, J.: Finite-difference solution of one-dimensional Stefan problem with periodic boundary conditions, Int. J. Heat mass trans. 46, 2911-2916 (2003) · Zbl 1041.80004 · doi:10.1016/S0017-9310(03)00050-4
[7]Kutluay, S.; Bahadir, A. R.; Ozdes, A.: The numerical solution of one-phase classical Stefan problem, J. comput. Appl. math. 81, 135-144 (1997) · Zbl 0885.65102 · doi:10.1016/S0377-0427(97)00034-4
[8]Meek, P. C.; Norbury, J.: Nonlinear moving boundary problems and a Keller box scheme, SIAM J. Numer. anal. 21, No. 5, 883-893 (1984) · Zbl 0558.65087 · doi:10.1137/0721057
[9]Liu, F.; Mcelwain, D. L. S.: A computationally efficient solution technique for moving-boundary problems in finite media, IMA J. Appl. math. 59, 71-84 (1997) · Zbl 0958.80007 · doi:10.1093/imamat/59.1.71
[10]Rizwan-Uddin: A nodal method for phase change moving boundary problems, Int. J. Comput. fluid dyn. 11, 211-221 (1999) · Zbl 0962.76074 · doi:10.1080/10618569908940875
[11]Caldwell, J.; Chiu, C. K.: Numerical solution of one-phase Stefan problems by the heat balance integral method, part II – special small time starting procedure, Commun. numer. Meth. eng. 16, 585-593 (2000) · Zbl 0964.65110 · doi:10.1002/1099-0887(200008)16:8<585::AID-CNM363>3.0.CO;2-7
[12]Caldwell, J.; Chiu, C. K.: Numerical solution of one-phase Stefan problems by the heat balance integral method, part I – cylindrical and spherical geometries, Commun. numer. Meth. eng. 16, 569-583 (2000) · Zbl 0964.65109 · doi:10.1002/1099-0887(200008)16:8<569::AID-CNM361>3.0.CO;2-3
[13]Goodman, T. R.: The heat-halance integral and its application to problems involving a change of phase, Trans. ASME 80, 335-342 (1958)
[14]Mitchell, S. L.; Myers, T. G.: A heat balance integral method for one-dimensional finite ablation, AIAA J. Thermophys. 22, No. 3, 508-514 (2008)
[15]Mitchell, S. L.; Myers, T. G.: Approximate solution methods for one-dimensional solidification from an incoming fluid, Appl. math. Comput. 202, No. 1, 311-5317 (2008) · Zbl 1208.76142 · doi:10.1016/j.amc.2008.02.031
[16]Myers, T. G.; Mitchell, S. L.; Muchatibaya, G.; Myers, M. Y.: A cubic heat balance integral method for one-dimensional melting of a finite thickness layer, Int. J. Heat mass trans. 50, 5305-5317 (2007) · Zbl 1140.80389 · doi:10.1016/j.ijheatmasstransfer.2007.06.014
[17]Myers, T. G.: Optimizing the exponent in the heat balance and refined integral methods, Int. commun. Heat mass trans. 36, No. 2, 143-147 (2009)
[18]S.L. Mitchell, T.G. Myers, Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Review, in press. · Zbl 1188.80004 · doi:10.1137/080733036
[19]S.L. Mitchell, T.G. Myers, Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions, IJHMT, in press. · Zbl 1194.80053 · doi:10.1016/j.ijheatmasstransfer.2010.04.015
[20]Rizwan-Uddin: One-dimensional phase change with periodic boundary conditions, Numer. heat trans. A 35, 361-372 (1999)
[21]Ascher, U. M.; Mclachlan, R. I.: Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. numer. Math. 48, 255-269 (2004) · Zbl 1038.65138 · doi:10.1016/j.apnum.2003.09.002
[22]Mitchell, S. L.; Morton, K. W.; Spence, A.: Analysis of box schemes for reactive flow problems, SIAM J. Sci. comput. 27, No. 4, 1202-1223 (2006) · Zbl 1136.65332 · doi:10.1137/030601910
[23]Strikwerda, J. C.: Finite difference schemes and partial differential equations, (2004)
[24]Schwerdtfeger, K.; Sato, M.; Tacke, K. -H.: Stress formation in solidifying bodies. Solidification in a round continuous casting mold, Metall. mater. Trans. B 29B, 1057-1068 (1998)
[25]Vynnycky, M.: An asymptotic model for the formation and evolution of air gaps in vertical continuous casting, Proc. roy. Soc. A 465, 1617-1644 (2009) · Zbl 1186.74037 · doi:10.1098/rspa.2008.0467 · doi:http://rspa.royalsocietypublishing.org/content/vol465/issue2105/
[26]M. Vynnycky, Air gaps in vertical continuous casting in round moulds, J. Eng. Math., submitted for publication.