Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems

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.     

An accurate numerical solution for the transient heating of an evaporating spherical droplet

A recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is extended for the purposes of solving a moving boundary problem for the transient heating of an evaporating spherical droplet. The Keller box finite-difference scheme is used, in tandem with the so-called boundary immobilization method. An important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second-order accuracy in both time and space - an issue not previously discussed in relation to this widely-used scheme. In addition, we demonstrate that our solution is in close agreement with the solution obtained using an alternative numerical scheme that employs an analytic solution of the heat conduction equation inside the droplet, for which the droplet radius was assumed to be a piecewise linear function of time. The advantages of the new method are discussed.     

A meshless method for one-dimensional Stefan problems

The paper presents a new meshless numerical technique for solving one-dimensional problems with moving boundaries including the Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries. The numerical examples are presented and the results are compared with analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness.     

Two algorithms for two-phase Stefan type problems

In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.     

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.     

An analytical and numerical study of the Stefan problem with convection by means of an enthalpy method

In this paper, we consider a theoretical and numerical study of the Stefan problem with convection, described by the Navier–Stokes equations with no‐slip boundary conditions. The mathematical formulation adopted is based on the enthalpy method. The existence of a weak solution is proved in the bidimensional case. The numerical effectiveness of the model considered is confirmed by some numerical results. Copyright © 2001 John Wiley & Sons, Ltd.     

Some exact solutions to Stefan problems with fractional differential equations

Some exact solutions to the first, second and extended Stefan problems with fractional time derivative described in the Caputo sense are given by means of fractional Green's function and Wright function in this paper. By the aid of simple calculations, many results of differential equations of integer order can be obtained as special cases of the results given by this paper.     

一类Stefan问题的适定性

本文讨论在烧蚀缓慢的情况下发汗控制微分方程并证明了其解的适定性。     

Fourth-order finite-difference method for boundary value problems with two small parameters

We present a finite difference scheme for a class of linear singularly perturbed boundary value problems with two small parameters. The problem is discretized using a Bakhvalov-type mesh. It is proved under certain conditions that this scheme is fourth-order accurate and that its error does not increase when the perturbation parameter tends to zero. Numerical examples are presented which demonstrate computationally the fourth order of the method.     

The quasilinearization method for boundary value problems on time scales

In this paper, we apply the method of quasilinearization to a family of boundary value problems for second order dynamic equations −y^Δ∇+q(t)y=H(t,y) on time scales. The results include a variety of possible cases when H is either convex or a splitting of convex and concave parts and whether lower and upper solutions are of natural form or of natural coupled form.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

An accurate LOD scheme for two-dimensional parabolic problems

We propose a high order locally one-dimensional scheme for solving parabolic problems. The method is fourth-order in space and second-order in time, and provides a computationally efficient implicit scheme. It is shown through a discrete Fourier analysis that the method is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with other schemes.     

On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions

Refined integral heat balance is developed for Stefan problem with time-dependent temperature applied to exchange surface. The method is applied to phase change in the half-plane and ordinary differential equation is obtained for the solid/liquid interface. The results are compared to those obtained by heat balance integral, perturbation and numerical methods.     

Solving two-phase freezing Stefan problems: Stability and monotonicity

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 non-linear 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.     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

Evolution of 3-D crystals from supersaturated vapor with modified Stefan condition: Galerkin method approach

We examine the evolution of crystals in three dimensions. We assume that the Wulff shape is a prism with a hexagonal base. We include the Gibbs-Thomson law on the crystal surface and the so-called Stefan condition. We show local in time existence of solutions assuming that the initial crystal has admissible shape.     

A qualocation method for a unidimensional single phase semilinear Stefan problem

Based on straightening the free boundary, a qualocation methodis proposed and analysed for a single phase unidimensional Stefanproblem. This method may be considered as a discrete versionof the H¹-Galerkin method in which the discretization is achievedby approximating the integrals by a composite Gauss quadraturerule. Optimal error estimates are derived in L(W^j,), j = 0,1,and L (H^j), j = 0,1,2, norms for a semidiscrete scheme withoutany quasi-uniformity assumption on the finite element mesh.     

Applying the combined integral method to one-dimensional ablation

In this paper the combined integral method is applied to a simple one-dimensional ablation problem. One of the drawbacks of heat balance integral methods is how to choose the approximating function. It is common to use a polynomial form but even then it is not clear what the power of the highest order term should be. Previous studies have determined exponents either from exact solutions or from expansions valid over short time scales; neither approach is satisfactory nor very accurate for larger times. We combine the heat balance and refined integral methods to determine this exponent as part of the solution process, and conclude that it is in fact time-dependent in the ablation stage. From comparing the approximate solutions with numerical and exact analytical solutions whenever possible, we show that this new method greatly improves the accuracy on standard methods, without overcomplicating the method.