Numerical Solution of the Stefan's Problem

All of the most well-known numerical methods for solving the Stefan’s problem, as well as a new method developed by the author are considered for the purpose of choosing the most efficient of them from a perspective of accuracy and computational speed. A comparison is carried out on the results of solving the problem for the boundary motion of “ice-water” phase transition around the vertical well passing through the thickness of permafrost. The conclusions, which are distributed to other multidimensional and multi-front statements of the Stefan’s problem, are made. The mathematical model, the brief description of the considered numerical methods and the boundaries of their applicability are presented. The comparison shows the advantages and disadvantages of different methods. It is demonstrated that the use of the explicit scheme leads to a marked increase in computation time, the six-point symmetric scheme may have oscillating solution; therefore, the implicit scheme is the most preferred. It is concluded that the most efficient method for one-dimensional and one front Stefan’s problems is the method of catching the front in the grid node using the implicit scheme, and the most efficient method for multi-dimensional and multi-front Stefan’s problems is the enthalpy method using the implicit scheme, which has been developed by the author.

Asymptotics of Solutions of the Generalized Hutchinson’s Equation

We discuss the dynamics of the Hutchinson’s equation and its generalizations. An estimate of the global stability region of a positive steady state is obtained. The main results refer to existence, stability and asymptotics of a slow oscillating solution. New asymptotic methods are applied to a problem of dynamical properties of ODE system describing Belousov — Zhabotinsky reaction.

DISSIPATIVE TRIGONOMETRICALLY-FITTED METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTION

In this paper a dissipative trigonometrically-fitted two-step explicit hybrid method is developed. This method is based on a dissipative explicit two-step method developed recently by Papageorgiou, Tsitouras and Famelis.6 Numerical examples show that the procedure of trigonometrical fitting is the only way in one to produce efficient dissipative methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions.

ON THE CONSTRUCTION OF EFFICIENT METHODS FOR SECOND ORDER IVPS WITH OSCILLATING SOLUTION

In this paper we describe procedures for the construction of efficient methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions. Based on the described procedures we develop two simple and efficient multistep methods for the solution of the above problems. The first method is exponentially-fitted and trigonometrically-fitted and the second has a minimal phase-lag. Both methods are symmetric. Numerical results obtained for several well known problems show the efficiency of the new methods when they are compared with known methods in the literature.

ALMOST CLASSICAL CREATION OF A UNIVERSE

We study the problem of a 1 + 1 cord with a dynamical massless scalar field living in it, which separates a false vacuum and a conical region in a 2 + 1 space. A stable "particle-like" configuration can be found. Also, oscillating solutions exist which can tunnel to an expanding type solution. The most outstanding feature for these oscillating solution is that we do not need a singularity to create an infinite universe from them, and that an arbitrarily small tunneling is needed to achieve this. Possible consequences for similar processes, involving cosmic strings in 3 + 1 dimensions are discussed.

Homogenisation of transport kinetic equations with oscillating potentials

We consider the homogenisation of transport kinetic equations with a highly periodic oscillating external field. The external field, acting on the particles, consists of a sum of a field deriving from a periodic potential and a bounded periodic perturbation. For the profile function generated by the oscillating solution of the problem, we derive a kinetic model with transmission boundary conditions in the energy variable. In some cases, for example when the field is not perturbed, we deduce a transport kinetic equation with memory effect satisfied by the weak-* limit of the sequence of solutions.