Numerical solution for the one-phase Stefan problem by piecewise-constant approximation of the interface
type: Item
instance of: scholarly article
title: Numerical solution for the one-phase Stefan problem by piecewise-constant approximation of the interface
zbMATH Open document ID: 0699.65082
DOI: 10.1007/BF02250635
author: Renato Spigler
author: Flavio Sartoretto
published in: Computing
publication date: 1990-01-01T00:00:00Z
review text: The classical one-phase one-dimensional Stefan problem is numerically solved on rectangles, \(R_ j\), of increasing size controlled by the Stefan condition. This approach is based on a scheme introduced by \textit{E. di Benedetto} and the second author [Rend. Semin. Mat. Univ. Padova 69, 109-134 (1983; Zbl 0476.35084)]. The practical implementation rests on the representation via thermal potentials of the solution \(u^ j(x,t)\) to the heat equation in \(R_ j\). The quantity \(u^ j_ x(x_ j,j\Delta t)\) which determines the \((j+1)\)-th rectangle is evaluated analytically by solving explicitly an integral equation. The solution in \(R_{j+1}\) is then obtained by numerically evaluating a further integral expression. The algorithm is tested by solving two problems whose solution is explicitly known. Convergence, stability and convergence rate as \(\Delta\) \(x\to 0\), \(\Delta\) \(t\to 0\) have been tested and plots are shown.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 80A22
zbMATH DE Number: 4147451
zbMATH Keywords: boundary integral methods
zbMATH Keywords: front tracking of the free boundary
zbMATH Keywords: one-phase one-dimensional Stefan problem
zbMATH Keywords: Stefan condition
zbMATH Keywords: thermal potentials
zbMATH Keywords: heat equation
zbMATH Keywords: algorithm
zbMATH Keywords: stability
zbMATH Keywords: convergence rate
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4147451