Vasilios Alexiades

Super-time-stepping acceleration of explicit schemes for parabolic problems

The goal of the paper is to bring to the attention of the computational community a long overlooked, very simple, acceleration method that impressively speeds up explicit time-stepping schemes, at essentially no extra cost. The authors explain the basis of the method, namely stabilization via wisely chosen inner steps (stages), justify it for linear problems, and spell out how simple it is to incorporate in any explicit code for parabolic problems. Finally, we demonstrate its performance on the (linear) heat equation as well as on the (non-linear) classical Stefan problem, by comparing it with standard implicit schemes (employing SOR or Newton iterations). The results show that super-time-stepping is more efficient than the implicit schemes in that it runs at least as fast, it is of comparable or better accuracy, and it is, of course, much easier to program (and to parallelize for distributed computing).

Existence theorems for some nonlinear fourth order elliptic boundary value problems

The general outline of our method is to that of of the second author second order elliptic parabolic equations [S, 71. of our effort is in deriving estimates for linear equations which lead to for the FrCchet derivatives of the relevant nonlinear to which Section 2 devoted, appear to be sharp in to explained below, and feel are independent interest. are stated in and the existence theorems developed in 4. a concluding section, a comparison is made between our results and on the theory of and pseudomonotone operators. There is substantial overlap in classes of to which the results and those obtained by The latter generally allows much greater in the nonlinearity (without priori estimate being available) whereas our results allow for a certain amount of ‘nonmonotonicity’ in the nonlinear terms. Furthermore, here we obtain a constructive algorithm (a variation of Newton’s method) for the solution, and this is a large part of the motivation for this work. In fact, the second author has recently utilized the second order results in [5] to obtain numerical approximations by the finite element method in [2]. We anticipate that the present work can be similarly utilized.

On the thermodynamic theory of fluid interfaces: Infinite intervals, equilibrium solutions, and minimizers

Abstract We outline a thermodynamic theory for one-dimensional fluid interfaces and compare our findings with the classical results of the variational van der Waals-Cahn-Hilliard approach. After establishing necessary and sufficient conditions for their equivalence, we list all types of possible solutions giving the structure of the density profile in an infinite interval. Then we examine the stability of these solutions, strictly within a variational thermodynamic context and prove that transitions are minimizers, but reversals and oscillations are not. To the best of our knowledge, this is the first proof available for this old problem. It substantiates previous intuitive statements and makes rigorous certain mathematical assertions existing in the physical literature.

Free Boundary Problems in Solidification of Alloys

Multidimensional three-phase free boundary problems for semilinear diffusion equations are studied as models for the solidification (or melting) of alloys. The intermediate phase represents the “mushy zone” in which the freezing of the remaining liquid provides a heat generation effect. The conditions on the two interfaces are of Stefan-type on one of them and of fast-chemical-reaction-type on the other. The existence, uniqueness and regularity of appropriately defined weak solutions are established when either the temperature or the heat flux is prescribed on the fixed boundary.

A Singular Parabolic Initial-Boundary Value Problem in a Noncylindrical Domain

The well-posedness of the first Fourier problem for a class of singular parabolic equations in noncylindrical domain is established in an appropriate weighted Hilbert–Sobolev space. The Green’s function is also constructed by means of potentials, and it is shown that the generalized and classical solutions coincide when the data permit the latter to exist.

The initial velocity of the emerging free boundary in a two-phase Stefan problem with imposed flux

Consider the one-dimensional, two-phase Stefan problem with an imposed flux, in which the phase change front originates at the material boundary. We prove that, subject to suitable assumptions on the initial temperature and imposed flux, the initial speed of the phase change front is equal to the jump between the surface and initial fluxes at the boundary. Similarly, we prove that the front

Singular Problems in the Theory of Stress-Assisted Diffusion

X(t)

A SINGULAR FOURIER PROBLEM WITH NONLINEAR BOUNDARY CONDITION

is continuously differentiable in the closed interval

RESOLVING THE CONTROVERSY OVER TIN AND GALLIUM MELTING IN A RECTANGULAR CAVITY HEATED FROM THE SIDE

[0,t^ * ]

A reference solution for phase change with convection

for some