A variable time step Galerkin method for a one-dimensional Stefan problem

Asaithambi, N. S.

doi:10.1016/0096-3003(95)00329-0

Cited by 16 publications

(15 citation statements)

References 10 publications

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“…In this article, we will provide condition 0 < s 0 ≤ s(τ ), τ ∈ (0, T 0 ] for a given finite time period T 0 > 0. With this condition, the Landau-type transformation (2.3) was used for some free boundary problems in the previous articles (see [6][7][8][9][10][11][12][13][14][15], etc).…”

Section: Remark 22mentioning

confidence: 99%

“…A rigorous analysis will be performed in the future work. Some numerical experiments related to the front fixing methods were also taken in [13] and [14]. Remark 6.4.…”

Section: Fully Discrete Finite Element Schemesmentioning

confidence: 99%

“…Gastaldi in [12] considered the approach of Nitsche for a linear Stefan-like problem with a generalized free boundary condition. Based on the front fixing technique, Asaithambi in [13] studied a variable time-step approximation to a one-dimensional Stefan problem describing the evaporation of droplets. D'Ambra in [14] discussed numerical simulation of the growth of a crystal from its melt by using a front fixing method.…”

Section: Introductionmentioning

confidence: 99%

“…In the article, we propose and analyze an H 1 -Galerkin method for the general single-phase nonlinear Stefan system of water-head, concentration, and free boundary equations [i.e., Problem (P) (1.1)-(1.7)]. For treating the free boundary, a Landau type transformation is introduced to transform the problem into one with a fixed domain similarly as in the previous works (see, [6][7][8][9][10][11][12][13][14][15]). For overcoming the difficulty that the water-head boundary condition depends on the free boundary function s(τ ), a transformation for the water-head and concentration functions is given to transform the nonhomogeneous Dirichlet condition into a normal homogeneous one.…”

Section: Introductionmentioning

confidence: 99%

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Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Dong

Wang

Sharpley

2006

Numer. Methods Partial Differential Eq.

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In this article, we consider a single-phase coupled nonlinear Stefan problem of the water-head and concentration equations with nonlinear source and permeance terms and a Dirichlet boundary condition depending on the free-boundary function. The problem is very important in subsurface contaminant transport and remediation, seawater intrusion and control, and many other applications. While a Landau type transformation is introduced to immobilize the free boundary, a transformation for the water-head and concentration functions is defined to deal with the nonhomogeneous Dirichlet boundary condition, which depends on the free boundary function. An H 1 -finite element method for the problem is then proposed and analyzed. The existence of the approximation solution is established, and error estimates are obtained for both the semi-discrete schemes and the fully discrete schemes.

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Section: Remark 22mentioning

confidence: 99%

“…A rigorous analysis will be performed in the future work. Some numerical experiments related to the front fixing methods were also taken in [13] and [14]. Remark 6.4.…”

Section: Fully Discrete Finite Element Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Dong

Wang

Sharpley

2006

Numer. Methods Partial Differential Eq.

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“…Reviews on numerical methods for solving various one-dimensional Stefan problems were recently given by Caldwell and Kwan [5] and Javierre, Vuik, Vermolen, and van der Zwaag [6]. Other earlier related references which may be of interest here include Asaithambi [7,8], Crank [9], Kutluay, Bahadir, and Özdeş [10], and Lesaint and Touzani [11].…”

Section: Introductionmentioning

confidence: 99%

A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

Ang

2007

Numerical Methods Partial

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A numerical method based on an integro-differential formulation is proposed for solving a one-dimensional moving boundary Stefan problem involving heat conduction in a solid with phase change. Some specific test problems are solved using the proposed method. The numerical results obtained indicate that it can give accurate solutions and may offer an interesting and viable alternative to existing numerical methods for solving the Stefan problem.

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Solving two‐phase freezing Stefan problems: Stability and monotonicity

Piqueras

Jódar

2019

Math Methods in App Sciences

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The two-phase Stefan problems with phase formation and depletion are special cases of moving boundary problems with interest in science and industry. In this work, we study a solidification problem, introducing a front-fixing transformation. The resulting nonlinear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.

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A variable time step Galerkin method for a one-dimensional Stefan problem

Cited by 16 publications

References 10 publications

Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

Finite element approximations to one-phase nonlinear free boundary problem in groundwater contamination flow

A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

Solving two‐phase freezing Stefan problems: Stability and monotonicity

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