Volume 28, Issue 3

The paper is focused on various aspects of application and justification of the standard vector algorithm for statistical simulation of polarized radiation transfer. Since the corresponding estimators may have an infinite variance, we construct an ‘algorithm of ℓ- fold polarization’ free from this drawback. Modifications of the local estimator of polarized radiation intensity and a dual representation of the mean square estimator of a functional are also considered, which extends the scope of studying the variance finiteness condition.

In this paper we study problems related to the finiteness of the variance of weighted Monte Carlo estimators for solving a system of second-kind integral equations. Modifications of the finiteness criterion for the variance of a weighted scalar estimator are based on the construction of an appropriate system of linear integral equations with ‘majorant’ kernels. A majorant criterion of the finiteness of a vector weighted estimator is given. This criterion uses a comparison of the estimator with a scalar estimator obtained by the method of randomization. It is shown that for the corresponding randomized algorithm the variance of the scalar estimator and the mean simulation time for a single trajectory are bounded if the original solution is bounded. The ability to use branching to construct a vector estimator of the solution to a system of integral equations is also studied.

In this paper we study scattering phase functions in water-drop clouds for various distributions of droplet size and various conditions of glory, rainbow and corona formation, and discuss the hypothesis proposed by A. N. Nevzorov that a considerable amount of water in cold clouds can exist in a specific phase state with the refractive index ≈ 1.8 (so called A-water). Polarization and angular distributions are studied by the Monte Carlo method for radiation reflected by cloud layers with drops of water or hypothetical A-water taking into account multiple scattering. Computational results make it possible to develop procedures for analysis of microphysical structure of clouds and confirmation or disproof of the existence of A-water.

We present a method for numerical simulation of the two-phase water flooding problem on general polyhedral grids not aligned with permeability tensor K (K-nonorthogonal grids) and dynamic octree grids adapted to the front between the phases. The discretization is based on the cell-centered monotone finite volume (FV) method with the nonlinear two-point flux approximation (TPFA) applicable to general K-non-orthogonal polyhedral grids. We use fully implicit discretization in time to avoid the restriction on the time step caused by the minimal mesh size. In our numerical experiments we demonstrate the superiority of the nonlinear TPFA on a K-non-orthogonal grid over linear TPFA and considerable speed-up of the simulation on a dynamically adapted octree grid with minimal loss in accuracy compared to the simulation on a fine regular grid.

A minimization method for a quadratic functional with linear constraints is proposed. The method obtains a solution in a finite number of steps determined by the choice of the initial guess, and the number of operations at each step depends polynomially on the dimension of the problem. The method does not use any a priori information

A model of underground coal gasification (UCG) is proposed in the paper. In the process of UCG, coal burning at the interface (so-called ‘fire-face’) between the coal mass and the gas chamber (called fire drift) is transformed into a gas mixture. A certain part of gas is filtered through the coal mass and rock. This process is described by Darcy’s equation. The greater part of the gas formed on the moving fire-face enters the fire drift. The gas motion in this drift is described by a system of Navier-Stokes equations for a compressible viscous heat-conducting fluid taking into account chemical reactions.