Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems. (English) Zbl 1177.80078

Mitchell, S. L.; Vynnycky, M.

Appl. Math. Comput. 215, No. 4, 1609-1621 (2009).

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:

80A22	Stefan problems, phase changes, etc.
80M20	Finite difference methods applied to problems in thermodynamics and heat transfer

Keywords:

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

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