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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.

The problem of changing tree species in protective afforestation remains poorly covered. The dynamics of species’ change in poplar ( Populus L.) protective forest belt growing next to the forest crops of the Scotch pine ( Рinus sylvestris L.) is discussed in the article. A forest belt 450 m long and 10 m wide grows in the steppe zone near the village of Shigaikulbash, Buzdyaksky District of the Republic of Bashkortostan. Accounting plots are laid every 50 m. Reliable pine undergrowth in the forest belt, poplar undergrowth was not found on any of the accounting plots. Comparative characteristics of pine undergrowth (density, average age, stem height, and diameter) and the condition of poplar trees were compared for 2008 and 2020. It was revealed that there was a change in the design of the forest belt from openwork to dense, there is a partial shrinkage of poplar, in some areas there is an exit of pine undergrowth to the first tier. For both periods, the highest density of pine (over 1.5 thousand trees/ha) was detected at a distance of up to 200 meters from pine forest crops, the lowest density (no more than 0.4 thousand trees/ha) - in the center of the forest belt. In 2020, the category of small undergrowth was not found, most of it is large undergrowth, some of which have entered the reproductive phase. The drying of poplar trees is observed. This was facilitated by a complex of reasons: the complete absence of agrotechnical care in the forest belts, a drought that has been recurring for several years and the maximum age of trees in the conditions of the steppe. There is no poplar undergrowth along the entire length of the forest strip. A similar pattern is observed in neighboring forest belts, which allows to make a conclusions about the actual change of balsam poplar ( Populus balsamifera L.) to Scotch pine in protective forest belts, which should be accompanied by logging and agrotechnical tillage on the edges of forest belts, followed by the formation of a young pine tree stand of openwork design.

The results of the studies of the features of adaptation of triploid aspen regenerants obtained in vitro using a hydroponic system are discussed in the article. Triploid clones of aspen ( Populus tremula L.) are characterized by rapid growth, high wood quality and increased resistance to diseases and pests compared to diploid clones and can be used to create forest plantations. Use of clonal micropropagation is advisable to preserve the valuable gene pool of triploid forms of aspen, while the adaptation of plants to ex vitro conditions is the most difficult process. Growing plants using the hydroponic method has several advantages. Triploid aspen regenerants, in which the length of the green mass prevails over the root mass, have a normal development without signs of infection and a good turgor of the green mass when they are removed from the test tube during the adaptation. Triploid aspen regenerants with decapitation of the apical shoot have the ratio of the lengths of the green mass to the root mass is close to 1:1, while the predominance of roots with values above the average is observed. Aspen plants with a predominance of root mass over green are formed from regenerants with leaving of 2 internodes and have an average of 2 roots of the first order. Triploid aspen plants-regenerants with biometric parameters of roots 5-7 cm and green mass up to 5 cm had the highest survival rates (94.7-100 %). A gradual decrease in the leaf surface area contributes to a better survival of regenerants, while it is necessary to reduce the humidity level from 100 % at the initial stage of adaptation to 60 % within 5 days, then - on 20 % with an interval of 3 days. The period of adaptation of triploid aspen plants in a hydroponic system is 16 days.

Species of the genus Cypress trees ( Chamaecyparis Spach), family Cupressaceae - monoecious evergreen trees with a cone-shaped dense crown and aromatic resinous needles. In nature, they grow in East Asia (China, Japan, Taiwan) and North America; ornamental plants suitable for urban gardening and modern urban floristry. These plants look good in single, group and avenue plantings, especially in places with a suitable climate. They give valuable light, fine-grained, fragrant, hard and durable wood. The varietal wealth is so great - only 4 species have produced over 1500 modern cultivars. In culture, many varieties of Japanese selection are known, which have already proven themselves well in other countries. In most cases, cypress trees are suitable in areas with a maritime climate, mild winters and high humidity. Comparison with literature and archival data shows that in recent years and decades, against the background of climate warming, there has been a significant increase in the size of plants. Blunt cypress ( Chamaecyparis obtusa (Siebold et Zucc.) Endl.), in the Peter the Great Botanical Garden of V. L. Komarov Botanical Institute, Russian Academy of Sciences has been known since 1870, it grown in the collection in different periods -the first mention from 1870 to 1898, then in the period from 1949 to 1977. It has been included in the modern collection of the arboretum since 2009. At the age of 14, it is a double-trunked tree 1.94 m high, with a trunk of 0.20 m. In 2018, seed production was observed for the first time. In 2021, seed offspring were obtained. Given the ongoing warming of the climate in St. Petersburg and increasingly favorable wintering conditions, blunt cypress has prospects for wider cultivation in the North-West of Russia and introduction into the modern range of gardens and parks in St. Petersburg as a new ornamental plant. For planting blunt cypress, one should choose a place protected from cold winds, without stagnant moisture, near groundwater, but a slightly shaded place is possible. In the summer, plants should be watered, this is especially important for young plants, preventing their earthen coma from drying out. In winter, during heavy snowfalls, it is useful to shake off the plants to avoid snow breakage.