A fixed-grid local method for 1-D Stefan problems

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

We treat two related moving boundary problems. The first is the ill-posed Stefan problem for melting a superheated solid in one Cartesian coordinate. Mathematically, this is the same problem as that for freezing a supercooled liquid, with applications to crystal growth. By applying a front-fixing technique with finite differences, we reproduce existing numerical results, concentrating on solutions that break down in finite time. This sort of finite time blow-up is characterised by the speed of the moving boundary becoming unbounded in the blow-up limit. The second problem, which is an extension of the first, is proposed to simulate aspects of a particular two phase Stefan problem with surface tension. We study this novel moving boundary problem numerically, and provide results that support the hypothesis that it exhibits a similar type of finite time blow-up as the more complicated two phase problem. The results are unusual in the sense that it appears the addition of surface tension transforms a well-posed problem into an ill-posed one. References S. D. Howison, J. R. Ockendon, and A. A. Lacey. Singularity development in moving-boundary problems. Q. J. Mech. Appl. Math., 38(3):343--360, 1985. doi:10.1093/qjmam/38.3.343 A. A. Lacey and J. R. Ockendon. Ill-posed free boundary problems. Control and Cybernetics, 14:275--296, 1985. A. Fasano and M. Primicerio. General free-boundary problems for the heat equation I. J. Math. Anal. Appl, 57:694--723, 1977. A. Fasano, M. Primicerio, and A. A. Lacey. New results on some classical parabolic free-boundary problems. Quart. Appl. Math., 38:439--460, 1981. M. A. Herrero and J. J. L. Velazquez. Singularity formation in the one-dimensional supercooled Stefan problem. Euro. J. Appl. Math., 7:119--150, 1996. doi:10.1137/04060528X B. Sherman. A general one-phase Stefan problem. Quart. Appl. Math., 28:377--382, 1970. J. R. King and J. D. Evans. Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem. SIAM J. Appl. Math., 65(5):1677--1707, 2005. doi:10.1137/04060528X J. Crank. Free and moving boundary problems. Clarendon Press, Oxford, 1984. T. C. Illingworth and I. O. Golosnoy. Numerical solutions of diffusion-controlled moving boundary problems which conserve solute. J. Comp. Phys., 209:207--225, 2005. doi:10.1016/j.jcp.2005.02.031 F. Liu and D. L. S. McElwain. A computationally efficient solution technique for moving-boundary problems in finite media. IMA J. Appl. Math, 59:71--84, 1997. doi:10.1093/imamat/59.1.71 S. W. McCue, B. Wu, and J. M. Hill. Classical two-phase Stefan problem for spheres. Proc. R. Soc. A, 464:2055--2076, 2008. doi:10.1098/rspa.2007.0315 S. W. McCue, B. Wu, and J. M. Hill. Micro/nanoparticle melting with spherical symmetry and surface tension. IMA J. Appl. Math., 74:439--457, 2009. doi:10.1093/imamat/hxn038 B. Wu, S. W. McCue, P. Tillman, and J. M. Hill. Single phase limit for melting nanoparticles. Appl. Math. Model., 33:2349--2367, 2009. doi:10.1016/j.apm.2008.07.009 H. G. Landau. Heat conduction in a melting solid. Quart. J. Appl. Math., 8:81--94, 1950. L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAM Journal on Scientific Computing, 18:1--22, 1997. S. C. Gupta. The classical Stefan problem: Basic concepts, modelling and analysis. Elsevier, Amsterdam, 2003.

Determination of one or two unknown thermal coefficients of a semi-infinite material through a two-phase stefan problem

An alternative two-phase formulation and analytical solution for a Stefan problem in one-dimension accounting for volume change and sensible heat

We consider the one-dimensional one phase Stefan problems with temperature or thermal flux boundary conditions. Exact and closed-form solutions for these two free boundary problems are very limited and highly restricted to particular initial and boundary conditions, and so are the similarity solutions. In general, only numerical methods allow one to completely solve such nonlinear problems. In this paper, the quasi-similarity solutions are defined and, for each problem, an analytical approximation of the interfacial position is given as function of the time-dependent boundary datum.

One-dimensional Shape Memory Alloy Problem with Duhem Type of Hysteresis Operator.- Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions.- Finite Time Localized Solutions of Fluid Problems with Anisotropic Dissipation.- Parabolic Equations with Anisotropic Nonstandard Growth Conditions.- Parabolic Systems with the Unknown Dependent Constraints Arising in Phase Transitions.- The N-membranes Problem with Neumann Type Boundary Condition.- Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport.- Asymptotic Properties of the Nitzberg-Mumford Variational Model for Segmentation with Depth.- The ?-Laplacian First Eigenvalue Problem.- Comparison of Two Algorithms to Solve the Fixed-strike Amerasian Options Pricing Problem.- Nonlinear Diffusion Models for Self-gravitating Particles.- Existence, Uniqueness and an Explicit Solution for a One-Phase Stefan Problem for a Non-classical Heat Equation.- Dislocation Dynamics: a Non-local Moving Boundary.- Bermudean Approximation of the Free Boundary Associated with an American Option.- Steady-state Bingham Flow with Temperature Dependent Nonlocal Parameters and Friction.- Some P.D.E.s with Hysteresis.- Embedding Theorem for Phase Field Equation with Convection.- A Dynamic Boundary Value Problem Arising in the Ecology of Mangroves.- Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method.- Finite Difference Schemes for Incompressible Flows on Fully Adaptive Grids.- Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation.- Capillary Mediated Melting of Ellipsoidal Needle Crystals.- Boundary Regularity at {t = 0} for a Singular Free Boundary Problem.- Fast Reaction Limits and Liesegang Bands.- Numerical Modeling of Surfactant Effects in Interfacial Fluid Dynamics.- The Value of an American Basket Call with Dividends Increases with the Basket Volatility.- Mathematical Modelling of Nutrient-limited Tissue Growth.- Asymptotic Hysteresis Patterns in a Phase Separation Problem.- Obstacle Problems for Monotone Operators with Measure Data.- Piecewise Constant Level Set Method for Interface Problems.- Dynamics of a Moving Reaction Interface in a Concrete Wall.- Adaptive Finite Elements with High Aspect Ratio for Dendritic Growth of a Binary Alloy Including Fluid Flow Induced by Shrinkage.- A Free Boundary Problem for Nonlocal Damage Propagation in Diatomites.- Concentrating Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Nonlinearity.- Existence of Weak Solutions for the Mullins-Sekerka Flow.- Existence and Approximation Results for General Rate-independent Problems via a Variable Time-step Discretization Scheme.- Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach.- Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries.- Solvability for a PDE Model of Regional Economic Trend.- Surface Energies in Multi-phase Systems with Diffuse Phase Boundaries.- High-order Techniques for Calculating Surface Tension Forces.- Simulation of a Model of Tumors with Virus-therapy.- Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction.

Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].

Soil moisture plays an important role in water and energy balance in land-atmospheric interaction, but is impacted directly by the groundwater table. Dynamic variation of the groundwater tablecan be described mathematically by a moving boundary problem. In this paper, the moving boundary problem is reduced to a fixed boundary problem through a coordinate transformation. A new model of groundwater table simulation is developed using the mass-lumped finite element method and is coupled with the land surface model of Variable Infiltration Capacity (VIC). The simulation results show that the new model not only can simulate the groundwater table dynamically, but also can evade the choice of water table depth scale in computation with a low computation cost.

Multiplicity and stability analysis in a free boundary problem arising from a dissolution-growth process

The first known exact analytical solutions are derived for the free vibrations of sectorial thin plates having their radial edges simply supported, with arbitrary conditions along their circular edges. This requires satisfying: (1) the differential equation of motion, (2) boundary conditions along the radial and circular edges, and (3) proper regularity conditions at the vertex of the radial edges. The solution to the differential equation involves ordinary and modified Bessel functions of the first and second kinds, of non-integer order, and four constants of integration. Utilizing a careful limiting process, the regularity conditions are invoked to develop two equations of constraint among the four constants for sector angles exceeding 180 deg (re-entrant corners). Moment singularities for re-entrant corners are shown to be the same as the ones determined by Williams (1952) for statically loaded sectorial plates. Frequency determinants and equations are generated for circular boundaries which are clamped, simply-supported, or free. Nondimensional frequency parameters are presented for all three types of configurations for sector angles of 195, 210, 270, 330, and 360 deg (i.e., re-entrant corners).

This paper presents an exact analytical solution of one-dimensional transport of a contaminant undergoing advection, dispersion, sorption, and first-order decay, subject to unique boundary conditions, a first-order decaying contaminant concentration in time at the source with a constant concentration at infinity. This exact solution provides a method to study a new area of physical processes and to evaluate approximate methods and models that have been developed to analyze these problems. Recently two fields have developed interest in modeling physical processes that are best conceptualized using a decaying source boundary condition. These two fields are long-term modeling of recalcitrant non-aqueous phase liquid (NAPL) spills and radioactive waste disposal. Both of which can be approximated using a decaying source term. This paper briefly introduces some general field transport problems that can conceptually be described using a decaying source boundary condition. The paper then describes the governing equations, boundary and initial conditions, and the solution techniques used to develop the analytical solution. Next the analytic solution is provided and discussed along with some example cases. The results of the exact analytical solution are compared to another similar solution from the literature.

Introduction Using finite-difference methods for European options is relatively straightforward, as there is no possibility of early exercise. As we have seen, the possibility of early exercise may lead to free boundaries. The chief problem with free boundaries, from the point of view of numerical analysis, is that we do not know where they are. This makes it difficult to impose the free boundary conditions, since we have to determine where to impose them as part of the solution procedure. (Recall that in Chapter 8 we simply imposed the boundary conditions at fixed grid points.) There are two distinct strategies for the numerical solution of free boundary problems. One is to attempt to track the free boundary as part of the time-stepping process. In the context of valuation of American options this is not a particularly attractive method, as the free boundary conditions are both implicit – that is, they do not give a direct expression for the free boundary or its time derivatives. We simply note the existence of such methods here, and refer the reader to the literature for a discussion of various boundary tracking strategies for implicit free boundary problems. The other strategy is to attempt to find a transformation that reduces the problem to a fixed boundary problem from which the free boundary can be inferred afterwards . There are many transformations that do this, but we consider only the particularly elegant method involving the use of the linear complementarity formulation.

Exact solution of the multi-cracked Euler–Bernoulli column

This paper presents two exact explicit solutions for the three dimensional dual-phase lag (DLP) heat conduction equation, during the derivation of which the method of trial and error and the authors’ previous experiences are utilized. To the authors’ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.