The Stefan problem for the Fisher–KPP equation

We study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here.     

An accurate finite-difference method for ablation-type Stefan problems

A recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is extended for the purpose of solving one-phase ablation-type moving boundary problems; in tandem with the Keller box finite-difference scheme, the so-called boundary immobilization method is used. An important component of the work is the use of variable transformations that must be built into the numerical algorithm in order to preserve second-order accuracy in both time and space. The analysis also determines that the ablation front initially moves as the time raised to the power 3/2; hence, it evolves considerably more slowly than the phase-change front in the classical Stefan problem with isothermal cooling.     

Global Stability and Decay for the Classical Stefan Problem

The classical one‐phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time‐dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary. We establish a global‐in‐time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf‐type inequalities.© 2015 Wiley Periodicals, Inc.     

One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition

A finite difference method is used to solve a one-dimensional solidification problem with a periodic boundary condition prescribed at the bottom of the mold of finite thickness. The temperature distributions in the solidified shell and mold, the position of the moving freezing front, and its velocity are evaluated. Analytical results are obtained for the limiting cases and then compared with the numerical predictions to establish the validity of the model and the numerical approach. Interactive effects of the process parameters such as Stefan number of the solidified shell material, the mold thickness, the thermal conductivity and thermal diffusivity between the shell and mold materials on the evolution of the freezing front and its velocity are investigated in detail. The results show that the solidified materials with larger Stefan number grow slower than those with relatively smaller Stefan number. The impact of oscillating mold temperature boundary on the growth of shell thickness is particularly significant at earlier stages of the process and more pronounced for smaller Stefan numbers. Increasing mold thickness or thermal conductivity ratio between the shell and mold materials slows down the evolution of the shell thickness.     

A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

A bidimensional inverse Stefan problem: Identification of boundary value

We propose to regularize the bidimensional inverse Stefan problem that is to determine the boundary temperature u(x,0,t) in the liquid phase in a medium of water and melting ice. This ill-posed problem is regularized by means of a convolution equation and an error estimate in L²(R²) is obtained. Numerical results are given.     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

A diffusive Lotka–Volterra model with Robin boundary condition and sign-changing growth rates in time-periodic environment

In this paper, a Lotka–Volterra model with Robin and free boundary conditions is considered in the heterogeneous time-periodic environment. We mainly consider the changes of local growth rates of native and invasive species that might be negative in some large regions. We study the spreading–vanishing dichotomy. When vanishing occurs, a native species cannot spread successfully as time goes to infinity. However, for an invasive species, in the long run, either it will go extinct or converge to the unique positive solution of time-periodic boundary value problem of logistic equation. When spreading occurs, both native and invasive species have upper and lower bounds. We also obtain the criteria for spreading and vanishing, and estimate of the asymptotic spreading speed.     

Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II

We study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For simplicity, we assume that the environment and the solution are radially symmetric. In the special case of one space dimension and homogeneous environment, this free boundary problem was investigated in Du and Lin (2010) [10]. We prove that the spreading-vanishing dichotomy established in Du and Lin (2010) [10] still holds in the more general and ecologically realistic setting considered here. Moreover, when spreading occurs, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. When the environment is asymptotically homogeneous at infinity, these two bounds coincide. Our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension.     

Mathematical modeling and study of the process of solidification in metal casting

The process of melting and solidification in metal casting is considered. The process is modeled by a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the problem and its finite-difference approximation are given. A numerical algorithm is presented for solving the direct problem. The results are described and analyzed in detail. Primary attention is given to the evolution of the solidification front and to how it is affected by the parameters of the problem. Some of the results are illustrated by plots.     

Spreading and vanishing in a diffusive intraguild predation model with intraspecific competition and free boundary

This paper is concerned with the spreading and vanishing phenomena in a diffusive intraguild (IG) predation model with intraspecific competition and free boundary in one dimensional space. The main objective is to obtain the asymptotic behavior of spread of an invasive or new IG prey species via a free boundary. In two cases, we prove a spreading‐vanishing dichotomy for this model, specifically, the IG prey species either successfully spreads to infinity as t→∞ at the front and survives in the new environment or spreads within a bounded area and dies out in the long run. The long time behavior of (R,N,P) and criteria for spreading and vanishing are also obtained. And then, we estimate the asymptotic spreading speed of the free boundary when spreading happens. Besides, two numerical examples are given to illustrate the impacts of initial occupying area and expanding capability on the free boundary.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

在纵向自由薄膜凝固期内温度分布的数值分析

考虑到薄膜表面的热通量主要是来自辐射，因而采用一个依赖时间的拟二维拟线性扩散方程的Ｓｔｅｆａｎ问题（混合初边值问题）作为该问题的数学模型。用一种具有Ｃｒａｎｋ－Ｎｉｃｈｏｌｓｏｎ格式的无条件稳定的有限差分析来求解抛物型偏微分方程的定解问题。用追赶法求解离散化的三对角方程组，然后用预估校正法求解拟线性扩散方程，从而避免了示解非线性差分方程组，琥珀亚硝酸酯在纵向自由薄膜凝固期内的温度分布数值计算结果和     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

STUDY OF CRYSTAL GROWTH AND SOLUTE PRECIPITATION THROUGH FRONT TRACKING METHOD

Crystal growth and solute precipitation is a Stefan problem. It is a free boundary problem for a parabolic partial differential equation with a time-dependent phase interface. The velocity of the moving interface between solute and crystal is a local function. The dendritic structure of the crystal interface, which develops dynamically, requires high resolution of the interface geometry. These facts make the Lagrangian front tracking method well suited for the problem. In this paper, we introduce an upgraded version of the front tracking code and its associated algorithms for the numerical study of crystal formation. We compare our results with the smoothed particle hydrodynamics method （SPH） in terms of the crystal fractal dimension with its dependence on the Damkohler number and density ratio.     

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell–Cattaneo equation for heat conduction. The model is formulated in terms of a two-phase Stefan problem. We consider the cases of the temperature profile being continuous or having a jump across the solid–liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used. However, only the jump condition avoids the onset of unphysical “supersonic” melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition, is the most suitable for use in models of phase change based on the Maxwell–Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.     

A free boundary problem for the stokes system with contact lines

In this paper we study an evolution free boundary problem for the two-dimensional Stoltes system in the case in which the free boundary intersects the solid boundary of a container. We prove existence and uniqueness of solutionsfor this problem in suitable classes of functions and under suitable smallness conditions for the initial data. For these solutions the contact point moves with an uniform velocity with respect to the container.     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

Single phase limit for melting nanoparticles

The melting of a spherical or cylindrical nanoparticle is modelled as a Stefan problem by including the effects of surface tension through the Gibbs–Thomson condition. A one-phase moving boundary problem is derived from the general two-phase formulation in the singular limit of slow conduction in the solid phase, and the resulting equations are studied analytically in the limit of small time and large Stefan number. Further analytical approximations for the temperature distribution and the position of the solid–melt interface are found by applying an integral formulation together with an iterative scheme. All these analytical results are compared with numerical solutions obtained using a front-fixing method, and are shown to provide good approximations in various regimes. The inclusion of surface tension, which acts to decrease the melting temperature as the particle melts, is shown to accelerate the melting process. Unlike the classical one-phase Stefan problem without surface tension, the solid–melt interface exhibits blow-up at some critical radius of the particle (which for metals is of the order of a few nanometres), a phenomenon that has been observed experimentally. An interesting feature of the model is the prediction that surface tension drives superheating in the solid particle before blow-up occurs.