Volume 32, Issue 6

A regularized system of equations describing a flow of isothermal two-component mixture with diffuse interface is studied. The equation of energy balance and its corollary, i.e., the law of non-increasing of the total energy are derived under general assumptions on the Helmholtz free energy of the mixture. Necessary and sufficient conditions for linearized stability of constant solutions are obtained (in particular case). A difference approximation of the problem is constructed in the two-dimensional periodic case on a nonuniform rectangular grid. The results of numerical experiments demonstrate a qualitative well-posedness of the problem and the applicability of the criterion of linearized stabilization in the original nonlinear formulation.

The issues of finite computational cost of some vector weighted Monte Carlo algorithms are studied in the paper relative to estimation of linear functionals of solutions to systems of the 2nd kind integral equations. A universal modification of the weight vector collision estimator with branching of the chain trajectory relative to the elements of matrix weight is constructed. It is proved that the computational cost of the constructed algorithm is finite in the case when the basic functionals are bounded. The results of numerical calculations are presented for the case of use of a modified weight estimator for some problems of the radiation transfer theory with allowance for polarization.

A problem of scattering of a monochromatic electromagnetic wave by a homogeneous dielectric body is considered. A part of the boundary of the body is a perfectly conducting thin surface. The problem is reduced to a system of the boundary integral equations containing integrals with strong singularity, the integrals are understood in terms of Hadamard final part. A numerical solution scheme is constructed on the base of approximate solution of these equations using the methods of piecewise-constant approximations and collocation. The constructed numerical scheme is tested on a model example.

The paper presents the results of application of a grid-characteristic method to simulation of wave processes in media with linearly-elastic and acoustic layers on the example of seismic exploration of the Arctic shelf. The grid-characteristic method correctly describes the contact and boundary conditions. The results of numerical simulation are presented, the method is shown to be applicable to solution of seismic exploration problem on the Arctic shelf.