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This article is devoted to a study of the properties of an implicit method for Abel type integral algebraic equations. An Abel type integral equation with stiff components is used for examining the properties of these methods and the stability domains are constructed. Numerical calculations confirming the results obtained are performed. In this article, a fractional “stiff” problem is proposed to study the stability of the mathematical objects considered.

Numerical differentiation of functions with large gradients is investigated. It is assumed that a function contains a component known up to a factor and responsible for the large gradients of the function. Application of classical formulas for calculating derivatives to such functions may lead to significant errors. Special-purpose formulas are studied for numerical differentiation on a uniform grid which are exact for a boundary layer component. Conditions are formulated under which an error estimate of a difference formula for a derivative does not depend on the gradients of the boundary layer component. In the case of an exponential boundary layer, when calculating a derivative of an arbitrarily given order error estimates that are uniform with respect to a small parameter are obtained. The results of numerical experiments are presented.

For optimal estimation of quantities of interest from observational data and a model (optimal filtering problem) in the nonlinear case, a particle method based on a Bayesian approach can be used. A disadvantage of the classical particle filter is that the observations are used only to find the weight coefficients with which the sum of the particles is calculated when determining an estimate. The present article considers an approach to solving the problem of nonlinear filtering which uses a representation of the posterior distribution density of the quantity being estimated as a sum with weights of Gaussian distribution densities. It is well-known from filtration theory that if a distribution density is a sum with weights of Gaussian functions, the optimal estimate will be a sum with weights of estimates calculated by the Kalman filter formulas. The present article proposes a method for solving the problem of nonlinear filtering based on this approach. An ensemble π-algorithm proposed earlier by the author is used to implement the method. The ensemble π-algorithm in this new method is used to obtain an ensemble corresponding to the distribution density at the analysis step. This is a stochastic ensemble Kalman filter which is local as well. Therefore, it can be used in high-dimensional geophysical models.

The paper is devoted to the mixed finite element method for the equation ΔΔυ + κ²υ = ƒ, x ∈ Ω, with boundary conditions υ = ϑυ/ϑν = 0 on ϑΩ, where ν is the normal to the boundary and κ ≥ 0 is an arbitrary constant on each finite element. At κ ≡ 0 residual type a posteriori error bounds for the mixed Ciarlet-Raviart method were derived by several authors at the use of different error norms. The bounds, termed sometimes a posteriori functional error majorants, seem to be less dependent on the constants in the general approximation bounds and are more flexible and adaptable for attaining higher accuracy at practical implementation. In this paper, we present a posteriori functional error majorants for the mixed Ciarlet-Raviart method in the case of κ ≠ 0 and having large jumps. Robustness and sharpness of the bounds are approved by the lower bounds of local efficiency.

A construction of the original HLL method for solving problems of relativistic hydrodynamics by using a piecewise-parabolic reconstruction of the physical variables is described. The resulting numerical method makes it possible to reproduce the numerical solutions with small dissipation at the discontinuities. The method is verified in problems of discontinuity breakdown in one-dimensional and two-dimensional formulation. The accuracy of the numerical scheme is studied in one-dimensional discontinuity breakdown problems. The method is also tested in typical astrophysical problems: interaction of relativistic jets, collision of clouds at relativistic speeds, and supernova explosion.

A two-dimensional eikonal equation which describes the propagation of tsunami wave fronts is considered. The paper presents the spatial form of an initial source of waves which causes accumulation of wave energy along the coastline of a round island. The theoretical results are confirmed by numerical simulation with a shallow water model of tsunami wave dynamics.

A realization of an adaptation criterion in the technology of generation of three-dimensional structured grids designed for the numerical solution of differential equations modeling the vortex processes of multi-component hydrodynamics is described. Earlier the adaptation criterion was realized for volumes of revolution and volumes of revolution deformed by other volumes of revolution. The adaptation criterion is realized within a variational approach for the construction of optimal grids satisfying optimality criteria: closeness of the grid to a uniform and orthogonal one and adaptation to a given function. In the realization of the criterion, the technology is supplemented by a new way of boundary nodes computation and an algorithm for the construction of an admissible set for minimization of a discrete functional formalizing the optimality criteria. Examples of grids adapted to a given function and its first derivatives are given.

The article deals with a boundary value problem of mathematical physics describing the unsteady process of diffusion of a reacting gas to a spherical coal particle located in the atmosphere of a mine. The solution of the boundary value problem is based on self-similar transformations which are a special case of group analysis. Formulas for determining the concentrations of the reacting gas in the vicinity of the coal particle and on its surface are obtained. Graphs of the dependencies of the burn-out time of the coal particle on a number of its parameters are constructed, and the fields of the reacting gas concentration at various stages of combustion of the coal particle are revealed.

The paper considers the urgency of the problem of flammability of sites for the location of oil and gas industry facilities and the need for pyrological expertise. For the first time in the Russian Federation, it is proposed to supplement the assessment of the impact on the environment of oil and gas complexes with the technology of pyrological expertise, developed at V. N. Sukachev Institute of Forest, Russian Academy of Sciences, Siberian Branch, and included in the Top-1000 list of the forum “Strong ideas for the new time-2022”. A proposal was made to supplement Clause 7.13.1.19 of the Order of the Ministry of Natural Resources dated December 1, 2020 № 999 “On approval of requirements for environmental impact assessment materials” with a fire-fighting arrangement of the territory, in accordance with the recommendations of the pyrological expertise. The concept of the pyrological expertise is given. The state of the issue on the assessment of natural fire danger in Russia and on the environmental policy of oil and gas companies is considered. To reduce the flammability of oil and gas fields, improve the fire safety of oil and gas facilities, better integrate Russian oil and gas companies into the energy transition process outlined by the Paris Agreement in 2015, and meet the requirements for industry decarbonization and ESG development, a complete review of corporate governance principles and available technologies is required, as well as shifting in mindset. The need to include vegetation fires, the number of which increases with the development of deposits, in the list of sources of greenhouse gases is noted. A review of materials and methods that can be used in the pyrological expertise is given. Guidelines have been developed for conducting the pyrological expertise based on a modern method for assessing fire hazard: making vegetation fuel maps and assessing the fire hazard of oil and gas facilities using the results of the Yurubcheno-Tokhomskoe field study as an example. The methodological recommendations presented in this article are based on many years of fundamental pyrological studies and the use of contemporary data for remote sensing of the Еarth and geoinformation technologies. Carrying out a pyrological expertise by oil and gas companies will help reduce vegetation fires, and hence reduce greenhouse gas emissions, increase the investment attractiveness of companies and the ESG-rating.

Significant areas of land has been withdrawn from the forest fund for the extraction of precious metals in the Urals. After the completion of the development of alluvial deposits, these lands require reclamation work. Low content in soils formed during the extraction of precious metals (gold, platinum, silver) of mineral nutrition elements and nitrogen excluder the agricultural direction of reclamation without the introduction of significant dozes of organic and mineral fertilizers. At the same time, the forestry direction of reclamation is effective by creating Scots pine ( Pinus sylvestris L.) forest crops. It has been established that the creation of forest crops by sowing causes their transfer to lands covered with forest vegetation’s with a density of 3.9 to 6.4 thousand pieces per ha of pine undergrowth. The proportion of decommissioned dead forest crops is 10.9 % of those created by this method. When creating forest crops by mechanized planting of 2-3 year old seedlings of Scots pine, the share of decommissioned forest crops is 4.3 % and in manually planting of the same age seedlings decommission of forest crops was not recorded. Due to the mosaic nature of soils, the transfer of established forest crops covered with forest vegetation of the ground varies from 4 to 11 years. An increase in the preservation of forest crops can ensured by the use first class quality seeds of local origin, since the dead forest crops were created mainly by seeds of the second quality class.