Abstract
The aim of the work is to build a software package for the distributed solution of the problem of matter transfer in a reservoir. A parallel implementation of the grid domain decomposition ...methods for computationally complex diffusion-convection problems, taking into account the architecture and parameters of a multiprocessor computing system located on the basis of the infrastructure facility of the University of Science and Technology «Sirius», is considered. An algorithm for parallel solution of the formulated hydrophysics problem using MPI technology has been developed. The analysis of the efficiency of the developed parallel algorithms is carried out and the study of the obtained solutions of computationally complex problems of aquatic ecology is carried out.
Abstract
The paper deals with the possibilities and prospects of experimental modeling of the interaction of a solid and a gaseous body. It is assumed that reliable experimental data have already ...been obtained and published. The problem of approximating the complex aerodynamic characteristics of air flow around a spherical body is posed and solved. The study was carried out using the «Cut-Glue» method for approximating numerical information about blowing experiments. Generally, this information reflects the dependence of the drag coefficient on the Reynolds number. The choice of the Cut-Glue method for the approximation of complex, multiextremal characteristics that can be obtained in physical experiments is substantiated.
Introduction. This work is aimed at solving the problem of phytoplankton dynamics in the coastal environments using the example of the Azov Sea. This takes into account the transformation of forms of ...phosphorus, nitrogen and silicon, as well as the aquatic medium motion, the distribution of temperatures and salinities over the sea area. River flow, varying in volume and chemical composition, affects significantly the variability of hydrophysical and biogeochemical parameters of the processes occurring in the coastal environment. This explains the need for statistical processing of the data from long-term observations over the river flow characteristics.
Materials and Methods. The mathematical model of biogeochemical cycles is based on a system of non-stationary equations of the convection-diffusion-reaction of parabolic type with nonlinear functions of sources and lower-order derivatives, to which the corresponding initial and boundary conditions are added. In the course of statistical analysis of the series of long-term observations over river flows, the values of the following indicators were found: skewness coefficient, degree of kurtosis, variance and standard deviation, coefficient of variation, autocorrelation coefficient, Neumann ratio, and Anderson criterion.
Results. The statistical analysis of the series of long-term observations over the hydrochemical indicators of the Don river suggests heterogeneity of the field data. This is due to the stochasticity of nutrient inputs and the volume of freshwater flow to the sea as a result of natural and anthropogenic factors. Field data should be correlated with seasonal changes in the aquatic environment temperature. This paper presents the results of a computational experiment to model the dynamics of phytoplankton populations in summer season, when temperatures are favorable for their reproduction and growth. The proposed mathematical model considers the spatially inhomogeneous distribution and transformation of forms of phosphorus, nitrogen, and silicon, as well as changes in salinity, temperature, and motion of the aquatic environment.
Discussion and Conclusions. The multispecies mathematical model of the dynamics of phytoplankton populations is considered with account for the transformation of forms of phosphorus, nitrogen, and silicon in the coastal environments. The analysis of data from field observations, for which its major statistical parameters are calculated, is carried out. As a result, it is concluded that data of the long-term observations are significantly variable. This is due to two reasons. Random nature of the input of nutrients and the volume of river flow as a result of anthropogenic factors is the first reason. The second reason includes the alternation of relatively high-water and low-water periods for fresh flow over the last 12-15 years. The hydrological regime is changing mainly due to the reduction of the average annual freshwater flow of the Don and partly of the Kuban. This trend is likely to increase due to climate changes, as well as with further regulation of the Don river flow after the Bagaevsky hydroelectric installation start-up. Numerical experiments based on the field data confirmed the predictive validity of the developed models and programs. They can be used to predict change in the composition and abundance (concentrations) in the Azov sea core planktonic populations, which define, on the one hand, food resources, and, on the other hand, the aquatic environment in terms of the ongoing sea salinization.
This paper is devoted to the construction and analysis of coupled mathematical models of hydrophysics and biological kinetics used for predicting hazardous natural phenomena occurring in shallow ...basins. The propagation and transformation of aquatic organisms is affected by such physical factors as three-dimensional spatial motion of water taking into account the advective transfer and microturbulent diffusion, spatially inhomogeneous distribution of temperature, salinity, and oxygen. Biogenic pollutants cause algae growth, including toxic and harmful ones; this growth can cause hazardous phenomena in the basin, including eutrophication and suffocation phenomena. A three-dimensional mathematical model of hydrodynamics is constructed and used for calculating the water flow velocity field. To investigate hazardous phenomena in a shallow basin related to suffocation phenomena in it, a three-dimensional spatially inhomogeneous ichthyological model of commercial fish dynamics is developed. Models of observations parameterized on the basis of stoichiometric relations, Monod, Michaelis–Menten, and Mitscherlich–Baule laws that describe the consumption and accumulation of nutrients by phytoplankton and commercial detritophagous fish and the growth of aquatic organisms depending on the spatial distribution of salinity, temperature, and oxygen regimen are considered. To calibrate and verify the models, constantly updated ecological databases obtained, in particular, in field research of the Sea of Azov and Taganrog Bay are used. To improve the accuracy of predictive simulation, the field data is filtered using the Kalman algorithm. As a result of processing the hydrological data, salinity and temperature isolines in the surface layer are obtained; for this purpose, a recognition algorithm is used. Using interpolation and superposition of domain boundaries, more detailed depth, salinity, and temperature maps for the Sea of Azov are obtained. Numerical methods for solving the formulated problems that are based on finite difference schemes taking into account the degree of filling of the computation domain control cells are developed. These methods are implemented on high-performance computers, and they decrease the numerical solution error and reduce the computation time by several fold. Based on the numerical implementation of the developed models, hazardous natural phenomena in shallow basins (related to the propagation of harmful pollutants), eutrophication, and algae bloom, which causes suffocation phenomena, are reconstructed.
In many countries of the world, there is a deterioration in the geoecological state of water bodies, associated with a significantly increased anthropogenic impact on natural waters. At the same ...time, the guaranteed quality of water resources and its reliability are ensured by comparing the realized quality and the guaranteed one. This article discusses models for the entry and movement of pollutants contained in the aquatic environment. Based on the currently used approaches and water pollution criteria, a set of parallel programs for high-performance computing systems has been developed, which allows modeling of the processes under consideration, as well as assessing risks and vulnerabilities in relation to anthropogenic impacts, zoning the water area of a shallow water body in accordance with the levels of anthropogenic loads, environmental design from the standpoint of sustainable development.
This paper discusses the numerical implementation of the mathematical model of the hydrodynamic process in the computational domain with an “extended geometry,” when its characteristic dimensions in ...the horizontal direction significantly exceed the vertical dimension. This is the typical property of a shallow water body or coastal system, which requires the development of specialized methods for solving the problems that arise in the process of discretizing grid equations. The explicit-implicit scheme has proved itself effective in solving the problem of transport in a shallow water body. The transition between time layers can be considered as an iterative process for solving the problem of diffusion-convection to settle. This idea forms the base for the formation of a preconditioner in the proposed method for solving grid equations obtained by approximating hydrodynamic problems in areas with extended geometry. A numerical experiment is carried out with the developed software module, which makes it possible to estimate the norm of the residual vector obtained by solving the grid equations of the pressure calculation problem based on the modified alternating triangular method (MATM) and the method for solving grid equations with a tridiagonal preconditioner, taking into account the hydrostatic approximation. According to the specifics of the developed method, it is effective in solving problems of aquatic ecology in the case of the computational domain, when its horizontal dimensions significantly exceed the vertical dimensions.
Abstract
The difference scheme is developed for the diffusion-convection problem numerical solution at large grid Peclet numbers. The scheme based on linear combination of the Upwind and Standard ...Leapfrog difference schemes (LCUSLDS) with weight coefficients obtained by minimizing the approximation error at small Courant numbers. The proposed approach takes into account the function of cell fullness, which allows to increase the accuracy of modeling on the area with complex geometry. The proposed modification of the Upwind Leapfrog difference scheme for the diffusion-convection problem numerical solution has better accuracy compared to other schemes considered in the article for grid values of Peclet numbers in the range 2 ≤ Pe ≤ 20. The problem of substances transport between two coaxial cylinders for large and small values of grid Peclet numbers is considered to test the proposed difference scheme. This movement of the water medium in hydrodynamics is called the Couette-Taylor flow. The steady fluid flow between two infinitely long coaxial circular cylinders is considered.
Introduction.
The solution to the problem of transformation of phosphorus, nitrogen and silicon forms is studied. This problem arises under modeling phytoplankton dynamics in shallow-water bodies ...including the Azov Sea. The phytoplankton dynamics model is formulated as a boundary value problem for the system of diffusion-convection-response equations and takes into account the absorption and release of nutrients by phytoplankton, as well as the transition of nutrients from one compound to another. To calculate the initial conditions and parameters of the equations under which the steady-state regime occurs, the software is developed, which is based on the model describing changes in phytoplankton concentrations without considering current effects. This model is represented by a system of inhomogeneous differential equations. Based on the developed software, the initial conditions and parameters of the phytoplankton dynamics model in the Azov Sea are calculated experimentally.
Materials and Methods.
A 3D model of phytoplankton dynamics is considered taking into account the transformation of phosphorus, nitrogen and silicon compounds based on the system of nutrient transport equations. The case of a spatially uniform distribution of substances is considered to specify the parameters of the model at which the stationary modes occur. Because of simplification, a system of ordinary differential equations solved through the Runge-Kutta method is obtained.
Research Results
. The software is developed to specify the initial conditions and parameters of the phytoplankton dynamics model considering the transformation of phosphorus, nitrogen and silicon compounds. Several numerical experiments are performed under the assumption that the development of phytoplankton is limited by a single biogenic substance. As a result of the computational experiment, it can be seen that with the obtained values of the initial concentrations and parameters of the equations, stationary modes occur for the system of ordinary differential equations describing the case of the spatially uniform distribution of substances.
Discussion and Conclusions.
The mathematical model of the transformation of phosphorus, nitrogen and silicon forms in the problem of phytoplankton dynamics is studied. Stationary modes for the system of ordinary differential equations are obtained, for which the values of the system parameters and initial conditions are determined. The results obtained can be used in further simulation of the phytoplankton dynamics considering the transformation of phosphorus, nitrogen and silicon compounds with account for convection-diffusion, salinity, and temperature.
The paper presents a three-dimensional hydrodynamic mathematical model that takes into account the processes of salt and heat transport in the Sea of Azov and makes it possible to obtain ...three-dimensional fields of water flow velocities, pressure, sea water density, salinity, and temperature. The model includes the Navier–Stokes equations of motion, the continuity equation in the case of variable density, and the equations of heat and salt transport. Boundary and initial conditions are specified. The diffusion–convection–reaction equation is approximated in time using weighted schemes. Approximation in spatial variables in the problem of determining the velocity field of the aquatic environment was performed by the balance method, taking into account the filling factors of the control regions. The initial distributions of salinity and temperature, which have a sufficient degree of smoothness at the points of setting their values, was performed using the Laplace equation. Maps of salinity and temperature of the Sea of Azov were obtained by interpolation and superimposing the boundaries of the region. Stationary regimes of heat and salt transfer have been studied. Based on the results of monitoring the water area, three-dimensional mathematical models of the movement of the aquatic environment were built, designed to predict possible scenarios for the development of the Azov Sea ecosystem, making it possible to study how anaerobic contamination areas emerge and take timely measures to localize them. To refine the input data of the mathematical model, an algorithm based on the Kalman filter was developed and implemented, which made it possible to obtain the minimum variance of the unbiased estimate of the state of the dynamic system. The elements of the software package are listed, making it possible to simulate hydrodynamic processes in shallow water bodies with complex spatial structures of currents with the transport of salts and heat.
Software package construction for a distributed solution of the matter transfer problem in a reservoir is the aim of the work. The software package consists of several components that use different ...methods of interaction. The solution study posed by various methods and comparison of their performance is carried out. An algorithm for parallel solution on a graphics accelerator controlled by the CUDA system has been developed. A comparative analysis of the algorithms operation on CPU and GPU is carried out. The software implementation of the components included in the complex is described; the main classes and implemented methods are documented.
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