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Zbl 1177.80078
Mitchell, S.L.; Vynnycky, M.
Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems. (English)
[J] Appl. Math. Comput. 215, No. 4, 1609-1621 (2009). ISSN 0096-3003

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 2000:: *80A22 Stefan problems, etc.
80M20 Finite difference methods

Keywords: Stefan problem; boundary immobilization; starting solutions; Keller box scheme; Crank-Nicolson scheme

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Zentralblatt MATH Berlin [Germany]