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The numerical algorithm without saturation for wave equation is considered. It is supposed that Laplace's operator has the discrete, valid range, and the corresponding matrix of the discrete operator Laplace has the complete set of eigenvectors. The technique speaks the example of the one-dimensional equation, but during statement is shown that the dimension is insignificant here.

The goal of this work is to highlight the advantages of using NonStandard Finite Differences (NSFD) numerical schemes for the resolution of Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution are a-priori known, such as positivity. The main reference considered is Mickens' work [14], in which the author derives NSFD schemes for ODEs and PDEs that describe real phenomena, and therefore widely used in applications. We rigorously demonstrate that NSFD methods can have a higher order of convergence than the related classical ones, deriving also the conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing the classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous.

In this paper, optimum differential schemes for the solution of the Maxwell equations with the use of the Laquerre spectral transformation are considered. Additional parameters are introduced into the differential scheme of equations for harmonics. Numerical values of these parameters are obtained by minimization of an error of differential approximation of the Helmholtz equation. The optimum values of parameters thus obtained are used when constructing differential schemes - optimum differential schemes. Two versions of optimum differential schemes are considered. It is shown that the use of optimum differential schemes leads to an increase in the accuracy of the solution of the equations. A simple modification of the differential scheme gives an increase in the efficiency of the algorithm.

This paper deals with Monte Carlo simulation of optical phenomena which appear in the lidar sensing of atmospheric clouds and water media. By numerical experiments we study peculiarities of the laser pulse propagation when the light forms expanding ring structures at the expense of multiple scattering.

In this paper, we, first, using the reduction method in the narrow sense (the simple reduction method), have generalized the classical Gauss-Jordan method for solving finite systems of linear algebraic equations to inhomogeneous infinite systems. The generalization is based on a new theory of solutions to inhomogeneous infinite systems, proposed by us, which gives an exact analytical solution in the form of a series. Second, we have shown that the application of reduction in the narrow sense in the case of homogeneous systems gives only a trivial solution, therefore, in order to generalize the Gauss-Jordan method for solving infinite homogeneous systems, we used the reduction method in the wide sense. A numerical comparison is given that shows acceptable accuracy.

The characteristics of the basic principles for solving the problems of interstate cooperation in transboundary river basins are presented. It is shown that in order to solve problems, it is necessary to have an idea of the patterns of functioning of these complexes under different economic policies of the states engaged in water use. Based on the long-term practices of interstate interaction, it was noted that the larger is the number of states intersecting a transboundary watercourse, the more difficult is the process of implementing measures of integrated water resources management. The main factors of development and exacerbation of water use conflicts in transboundary river basins are identified. An analysis of institutional and legal formats of interaction between states - water users has been carried out. The predominance of bilateral agreements and the lack of interest in the development of universal multilateral treaties were noted. From the standpoint of structural and functional integrity, the expediency of considering transboundary river basins from the standpoint of the basin principle is substantiated. It is concluded that under conditions of significant anthropogenic fragmentation of transboundary river basins, a quasi-integral natural-social geosystem of the regional level is formed. It is noted that taking into account the complexity and inconsistency of the “transboundary river basin” category, a universal approach is needed to solving the problems of interstate interaction, which makes it possible to combine the basic principles (limnological, institutional-legal and basin principle) into a single conceptual model. Such tasks can be successfully solved through the implementation of the main provisions of a systematic approach, the application of which will make it possible to determine the individual characteristics of a transboundary basin and synthesize them into a single whole, develop integrated water resource management schemes, coordinate joint actions of water-user states and move from private solutions of problems of interstate interaction to complex problems.

It has been defined that the clear spatial differentiation in the Far East is characteristic of the manifestation of the continentality and oceanicity interaction: minimal in the central areas of the continents and oceans and maximal in the transboundary ones. It has been established that the probability of origin and development of the related anomalous processes and disasters was included in the relation of continentality and oceanicity. Many peculiarities of the dynamic of the geosystems and anomalous processes were clarified. So, for example, the number and intensity of anomalous processes increase: the frequency of storms and storm surges grows up as well as the washway of the sea shores and submarine nearshore slope etc. intensifies. As a result, the strip of active interpenetation of the continentality and oceanicity emphasized currently by mottled alternation of the “strange” by origin forms, processes and formations (in the goltsy zone of the middle altitudes - formation of rock glaciers and cryogenic sorting of grounds and ice blisters and more in the valleys of mountain rivers) becomes displaced to the east, to the coastal zone. The repetition frequency of the extreme processes and their effect on the geosystems in the continental margin of the Russian Far East increase in recent decades. It has been established that the use of continentality and oceanicity characteristics allows us to determine the ecological risks and has the high potential of application in the solving of the urgent problems. It has been stated that the selection of the sustainable nature management strategy in the areas under consideration should make allowance for the existing risks and environmental restrains determined by them.

It has been established that the main factor hindering the development of agricultural production in Russia is soil degradation of agricultural landscapes, including chernozems - the country’s most important natural resource. It is concluded that the chief reason behind the deterioration of the genetic properties of soils is a long period of their extensive use. The changes in the political and economic systems of the country that took place at the turn of the 20^th and 21^st centuries also affected agriculture. A course was taken for the transition to intensive farming, for achieving maximum possible crop yields in small areas with high quality soils. To work out the methods of intensive farming within the chernozem belt of the Ural, it became necessary to preliminarily rehabilitate soil properties. In this regard, the study of the processes of restoration of the properties of degraded chernozems was the goal of this study. As a method of restoring soil properties, a largely forgotten agricultural method was used, which is a long-term fallow. The restoration of the main properties of chernozem was carried out simultaneously with the recovery of the species composition of vegetation and included three stages: the stage of “young” fallow (5 years from the time of withdrawal from arable land), “middle-aged”, or 12 years of stay in the fallow, and “old-age”, after a 25-year period of restoration of soil properties. At the end of each stage, the species composition of phytocenoses was studied and the chemical, biological and physical properties of chernozems were determined. It has been established that at the end of the period of stay of soils in the state of fallow, they almost completely restore their genetic properties, including those that ensure their fertility. Thus this agricultural method does not require capital expenditures for its implementation and allows the relevant goal to be achieved.

This paper presents results from determining the soluble and total forms of metals in natural environments of the Bazaikha river basin in the Krasnoyarsk agglomeration. It was found that the copper, aluminum and zinc concentrations in snow water filtrate exceeded the ecological standards for fishery waters. According to the increase in the proportion versus the background, the water-soluble forms of heavy metals in snow cover were arranged in the following series: Mn (1.14 mg/kg) < Cu (1.79) < Ni (2.38) < Al (3.25) < Fe (4.20) < Zn (4.46) < Sr (5.71 mg/kg). The content of mineral fraction in solid precipitate suggests a predominance of zinc-nickel-strontium aerotechnogenic pollution of snow cover as compared with the background. According to the increase in the mean concentration of metals, the mineral form of elements in solid precipitate were arranged in the following series: Cu (20.1 mg/kg) < Ni (46.8) < Sr (277.6) < Zn (386.6) < Mn (752.5) < Al (7457.7) < Fe (14 881.7 mg/kg). It was established that the concentration of copper and aluminum in the surface waters of the Bazaikha river exceeds ecological standards for fishery waters. In the order of increasing the mean content of water-soluble forms of metals in the surface waters of the river, the following sequence is obtained: Mn (0.001 mg/dm3) = Cu (0.001) < Sr (0.085) < Al (0.090) < Fe (0.091 mg/dm3), while in the order of increasing the mean concentrations of suspended forms of metals the sequence is arranged thus: Cu (0.002 mg/dm3) < < Mn (0.010) < Sr (0.132) < Fe (0.483) < Al (0.608 mg/dm3). It is concluded that most of the elements are transported by the river in the suspended runoff, which is characteristic for manganese, aluminum and iron. Almost 50 % of strontium and copper compounds in such redox conditions migrate in the water-soluble form. Geochemical coefficients as calculated from results of analyzing the collected samples show that in conditions of the Bazaikha rivershed there is a low degree of involvement of the metal masses under investigation in water migration processes.

Results from studying the flow of major ions in the water of the Amur river near Khabarovsk for the maximum water discharge in 2013 are presented. The effect of the regulated Zeya and Bureya rivers on the hydrological and hydrochemical regime of the Middle Amur is considered. An uneven distribution of the sum of ions across the width of the river was noted, with the greatest differences observed during the winter low-water period and the leas differences occurred during the spring high-water period. It was found that at the crest of the historical flood the differences in the distribution of the sum of ions across the width of the Amur did not exceed 26 mg/L, and the maximum value in the middle of the river (up to 82.8 mg/L) was higher than during the 1998, 2002, 2009 and 2019 floods. The yearly largest sum of ions (132 mg/L) was recorded as a result of the removal of dissolved matter from the flooded farmland and urbanized areas in December at the beginning of freeze-up. The priority influence of the Sungari river on the temporal and spatial variability in ion concentrations in the water of the Amur river is shown. An estimate was made of the ionic flow during the winter low-water period, spring flooding and historical flooding, the decline of which in September was accompanied by the greatest removal of salts (6644 thousand tons). Hydrocarbonate and sulfate ions, the calcium ion and, to a lesser extent, the potassium ion and chloride ion made the main contribution to the formation of the ionic flow. The maximum share of the ionic flow of anthropogenic genesis (sulfates and chlorides) is observed in May during spring flood, when snowmelt dominate in the Amur water runoff as well as of the hydrocarbonate ion and sodium ion in December, when salts are exported from moisture-saturated soils.