Vol 8, No 2 (2016)

This work is devoted to the study of the accuracy of the Lebesgue averaging method for the spectra of resonance radiation in solving the transport equation. The method is tested for the problem of the thermal radiation transfer in the Earth atmosphere. The results of numerical calculations in two versions of the Lebesgue averaging method are compared with the results of the direct pointwise (lineby-line) spectral calculations. In the first version, the classical Lebesgue integral is used, and the absorption coefficient is taken in it as the independent variable instead of the photons energy. In the second version, we apply the Lebesgue-Stieltjes integral with the measure of Lebesgue’s set as the independent variable. All calculations are performed by using one scheme of spatial and angular discretization of the transport equation. This ensures the purity of the computational experiment. The second variant of averaging by the use of Lebesgue-Stieltjes integration showed the highest accuracy (within 5%) with a significant decrease of the number of arithmetic operations (approximately by a factor of 10³–10⁴ ) in relation to the direct spectral calculations.

A new multigrid method is proposed for the solution of systems of linear algebraic equations obtained as a result of the discretization of the initial boundary-value problems for a heat equation with a discontinuous heat conduction coefficient. In the method, a special construction of the next level grid is used, with special treatment of subregions near the discontinuity lines of the heat conduction coefficient. The numerical experiments with a 2D model problem discretized on orthogonal grids demonstrated a high convergence rate for the method and weak dependence of the convergence on the discontinuity jump of the coefficient.

Possible approaches to modeling two-dimensional coherent hydrodynamic structures based on the statistical mechanics of local vortices are considered. The exact definitions of coherent structures are given and the mechanisms of their formation are shown. The bases of the kinetic theory of Onsager vortices are given and the possibility of applying the classical molecular-kinetic theory for the explanation of the origin of vortex meso-structures in the shear flows is considered.

Simple and robust formulas of the conjugate direction method for symmetric matrices and of the symmetrized conjugate gradient method for nonsymmetric matrices have been constructed. These methods were compared with robust forms of the conjugate gradient method and the Craig method using test problems. It is shown that stability for the round-off error can be attained when recurrent variants of the methods are used. The most reliable and efficient method for symmetric signdefinite and indefinite matrices appears to be the method of conjugate residuals. For nonsymmetric matrices, the best results have been obtained by the method of symmetrized conjugate gradients. These two methods are recommended for writing standard programs. A reliable criterion has also been constructed for the termination of the calculation on reaching background values due to the round-off errors.

A new approach to polynomial higher-order approximation (smoothing) based on the basic elements method (BEM) is proposed. A BEM polynomial of degree n is defined by four basic elements specified on a three-point grid: x₀ + α < x₀ < x₀ + β, αβ <0. Formulas for the calculation of coefficients of the polynomial model of order 12 were derived. These formulas depend on the interval length, continuous parameters α and β, and the values of f^(m)(x₀+ν), ν = α, β, 0, m = 0,3. The application of higher-degree BEM polynomials in piecewise-polynomial approximation and smoothing improves the stability and accuracy of calculations when the grid step is increased and reduces the computational complexity of the algorithms.

In the Reynolds equations with the Shear Stress Transport (SST) turbulence model, the numerical simulation of the effect of the energy input into the stream in front of the bow and the side surface on the base flow has been performed. For the regimes considered, it has been shown that the energy input before the bow, resulting in a significant reduction of wave resistance has little effect on the value of the base pressure. This ensures the efficiency of the energy input as a means of reducing the drag force. For the considered regimes, it has been shown that the input of energy around the lateral surface leads to a small increase in the base pressure.