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Technical Briefs

# A Four-Step Fixed-Grid Method for 1D Stefan Problems

[+] Author and Article Information
M. Tadi

Department of Mechanical Engineering, University of Colorado at Denver, Campus Box 112, P.O. Box 173364, Denver, CO 80217-3364mohsen.tadi@ucdenver.edu

J. Heat Transfer 132(11), 114502 (Aug 16, 2010) (4 pages) doi:10.1115/1.4002148 History: Received January 02, 2010; Revised July 07, 2010; Published August 16, 2010; Online August 16, 2010

## Abstract

This note is concerned with a fixed-grid finite difference method for the solution of one-dimensional free boundary problems. The method solves for the field variables and the location of the boundary in separate steps. As a result of this decoupling, the nonlinear part of the algorithm involves only a scalar unknown, which is the location of the moving boundary. A number of examples are used to study the applicability of the method. The method is particularly useful for moving boundary problems with various conditions at the front.

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Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 1

Finite difference mesh close to the interface

Figure 2

The region of stability for the explicit-implicit mesh at (k−2,n+1) just before the front for σ=1. The funtion is positive for λ≤2. The x axis is λΔx with the range [0,3.2]. The y axis is λ with the range [0,1.8]. The z axis is the function f(λ,ℓΔx).

Figure 3

The location of the front for example 1. The figure compares the exact solution to the numerical result.

Figure 4

The location of the front for example 2. The figure compares the exact solution to the numerical result. At t=2 the numerical solution is s(2)=0.4095, whereas the exact solution is s(2)=0.4107.

Figure 5

The location of the front for the example 3. The figure compares the exact solution to the numerical result. At t=2 the numerical solution is s(2)=0.8628, whereas the exact solution is s(2)=0.8614.

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