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The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method.

The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem.

The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat.

The paper presents a new solution algorithm for the Stefan problem.

A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.

The use of explicit finite difference schemes for low Stefan number problems with moving interface was largely abandoned because they require small time intervals (large CPU time) to obtain accurate non‐oscillatory solutions. This paper uses these type of schemes for better estimations of the dynamics of the solid—liquid interface. The scheme in which time and radial intervals are constant, uses a local, continuous, time‐dependent radial coordinate to define the instantaneous location of the interface. Taylor series expansions which result in a polynomial fit are used for forward and backward interpolation of temperatures of nodal points in the vicinity of the interface. A distinction is made between the left and right position of the interface relative to the closest nodal point. The algorithm handles accurately and effectively the non‐linearities near the interface thus producing accurate stable solutions with relatively low CPU time. The scheme which obviously may be applied to large Stefan number problems, is also suitable for time dependent boundary conditions as well as temperature dependent physical properties. The results obtained by the scheme were in excellent agreement with ones derived from an approximate analytical solution which is applicable in the low Stefan number range.

The purpose of this paper is to improve the accuracy and stability of the existing solutions to 1D Stefan problem with time-dependent Dirichlet boundary conditions. The accuracy improvement should come with respect to both temperature distribution and moving boundary location.

The variable space grid method based on mixed finite element/finite difference approach is applied on 1D Stefan problem with time-dependent Dirichlet boundary conditions describing melting process. The authors obtain the position of the moving boundary between two phases using finite differences, whereas finite element method is used to determine temperature distribution. In each time step, the positions of finite element nodes are updated according to the moving boundary, whereas the authors map the nodal temperatures with respect to the new mesh using interpolation techniques.

The authors found that computational results obtained by proposed approach exhibit good agreement with the exact solution. Moreover, the results for temperature distribution, moving boundary location and moving boundary speed are more accurate than those obtained by variable space grid method based on pure finite differences.

The authors’ approach clearly differs from the previous solutions in terms of methodology. While pure finite difference variable space grid method produces stable solution, the mixed finite element/finite difference variable space grid scheme is significantly more accurate, especially in case of high alpha. Slightly modified scheme has a potential to be applied to 2D and 3D Stefan problems.

The solidification of a superheated fluid‐porous medium contained in a rectangular cavity is studied numerically. The bottom and side walls of the cavity are insulated while the top wall is maintained at a constant temperature below the freezing point of the saturating fluid. The study is focused on the effects of superheat on the development of natural convection and heat transfer during the solidification process. For a fluid initially at a temperature above the freezing point, the results obtained by neglecting convection overpredicts the solidification time by about 12 percent for a Rayleigh number of 800. When convection is taken into account, it is found that the solidification process consists of three distinct regimes: the conduction regime, convection regime and the solidification of the remaining fluid that can be described by the Neumann solution for the solidification of a fluid at its freezing point. The numerical simulations are based on the Darcy‐Boussinesq equations, using the front tracking method in a transformed coordinate system. The entire solidification process is described in terms of the evolutions of the streamlines and isotherm patterns, the maximum and average temperatures of the fluid, the interface position, and the heat transfer rate. The parametric domain covered by these simulations is 0 ≤ Ra ≤ 800, 0 ≤ St^l ≤ 0.67, St^s = 0.3 and XL = 1 where Ra is the Rayleigh number, St^l the liquid Stefan number, St^s the solid Stefan number, and XL the aspect ratio of the cavity.

The purpose of this paper is to provide an insight and to solve numerically the identification of timewise terms and free boundaries coefficient appearing in the heat equation from over-determination conditions.

The formulated coefficient identification problem is inverse and ill-posed, and therefore, to obtain a stable solution, a nonlinear Tikhonov regularization least-squares approach is used. For the numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB subroutine lsqnonlin.

The numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.

The mathematical formulation is restricted to identify coefficients in unknown components dependent on time, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data is also limited to single temperature in heat equation with Stefan conditions, and the first- and second-order heat moments measurements at a particular time location.

As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.

The identification of the timewise terms and free boundaries will be of great interest in the heat transfer community and related fluid flow applications.

The current investigation advances previous studies, which assumed that the coefficient multiplying the lower order temperature term depends on time. The knowledge of this physical property coefficient is very important in heat transfer and fluid flow. The originality lies in the insight gained by performing for the numerical simulations of inversion to find the timewise terms and free boundaries coefficient dependent on time in the heat equation from noisy measurements.

Thermal solidification processes are an important concern in today’s manufacturing technology. Because of the complex geometric nature of real‐world problems, analytical techniques with closed‐form solutions are scarce and/or not feasible. As a consequence, various numerical techniques have been employed for the numerical simulations. Of interest in the present paper are thermal solidification problems involving single or multiple arbitary phases. In order to effectively handle such problems, the finite element method is employed in conjunction with adaptive time stepping approaches to accurately and effectively track the various phase fronts and describe the physics of phase front interactions and thermal behaviour. In conjunction with the enthalpy method which is employed to handle the latent heat release, a fixed‐grid finite element technique and an automatic time stepping approach which uses the norm of the temperature distribution differences between adjacent time step levels to control the error are employed with the scale of the norm being automatically selected. Several numerical examples, including single and multiple phase change problems, are described.

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

The homographic approximation, in which the Heaviside step function is replaced by a continuous smooth curve, is applied to the enthalpy method for heat transfer problems with isothermal phase change. Both the finite difference and finite element implementations, based on the basic enthalpy, the apparent heat capacity and the source term formulations, are considered. A 1‐D Stefan problem of melting a solid is used as a test problem. The accuracy of the numerical solutions is measured globally using L2 error norms and comparison is made between the solutions using homographic approximation and those using linear approximation. The advantages of using homographic approximation are examined.

To present a novel moving boundary problem related to the shoreline movement in a sedimentary basin and demonstrate that numerical techniques from heat transfer, in particular enthalpy methods, can be adapted to solve this problem.

The problem of interest involves tracking the movement (on a geological time scale) of the shoreline of a sedimentary ocean basin in response to sediment input, sediment transport (via diffusion), variable ocean base topography, and changing sea level. An analysis of this problem shows that it is a generalized Stefan melting problem; the distinctive feature, a latent heat term that can be a function of both space and time. In this light, the approach used in this work is to explore how previous analytical solutions and numerical tools developed for the classical Stefan melting problem (in particular fixed grid enthalpy methods) can be adapted to resolve the shoreline moving boundary problem.

For a particular one‐dimensional case, it is shown that the shoreline problem admits a similarity solution, similar to the well‐known Neumann solution of the Stefan problem. Through the definition of a compound variable (the sum of the fluvial sediment and ocean depths) a single domain‐governing equation, mimicking the enthalpy formulation of a one‐phase melting problem, is derived. This formulation is immediately suitable for numerical solution via an explicit time integration fixed grid enthalpy solution. This solution is verified by comparing with the analytical solution and a limiting geometric solution. Predictions for the shoreline movement in a constant depth ocean are compared with shoreline predictions from an ocean undergoing tectonic subsidence.

The immediate limitation in the work presented here is that “off‐shore” sediment transport is handled in by a “first order” approach. More sophisticated models that take a better accounting of “off shore” transport (e.g. erosion by wave motion) need to be developed.

There is a range of rich problems involving the evolution of the earth's surface. Many of the key transport processes are closely related to heat and mass transport. This paper illustrates that this similarity can be exploited to develop predictive models for earth surface processes. Such models are essential in understanding the formation of the earth's surface and could have a significant impact on natural resource (oil reserves) and land (river restoration) management.

For the most part the solution methods developed in this work are extensions of the standard numerical techniques used in heat transfer. The novelty of the work presented rests in the nature of the problems solved, not the method used. The particular novel feature is the time and space dependence of the latent heat function; a feature that leads to interesting analytical and numerical results.