Fig 5
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The paper is dedicated to the development and application of the 3D shallow water hydrodynamics model. Parallel realization
of the deepest descent the symmetric successive over relaxation (SSOR) algorithm is presented for solving finite-difference
equations obtained after discretization of the initial problem. Estimates for speeding up and increasi...
Citations
... With the development of computational fluid dynamics (CFD), it is widely used in flood simulation and prediction, groundwater transport simulation, and tidal movement analysis, of which basin-scale hydrodynamic simulation is indispensable for many engineering applications [1][2][3][4]. In current scenarios of basin-scale hydrodynamic simulation, the simulation area is usually a large-scale basin at the km level, and the simulation duration is usually set to a long time span, such as months or years [5,6]. The application of traditional models in basin-scale hydrodynamic prediction is limited due to factors such as huge computing scales, extended prediction spans, and lacking boundary conditions. ...
Traditional hydrodynamic models face the significant challenge of balancing the demands of long prediction spans and precise boundary conditions, large computational areas, and low computational costs when attempting to rapidly and accurately predict the nonlinear spatial and temporal characteristics of fluids at the basin scale. To tackle this obstacle, this study constructed a novel deep learning framework with a hydrodynamic model for the rapid spatiotemporal prediction of hydrodynamics at the basin scale, named U-Net-ConvLSTM. A validated high-fidelity hydrodynamic mechanistic model was utilized to build a 20-year hydrodynamic indicator dataset of the middle and lower reaches of the Han River for the training and validation of U-Net-ConvLSTM. The findings indicate that the R2 value of the model surpassed 0.99 when comparing the single-step prediction results with the target values. Additionally, the required computing time fell by 62.08% compared with the hydrodynamic model. The ablation tests demonstrate that the U-Net-ConvLSTM framework outperforms other frameworks in terms of accuracy for basin-scale hydrodynamic prediction. In the multi-step-ahead prediction scenarios, the prediction interval increased from 1 day to 5 days, while consistently maintaining an R2 value above 0.7, which demonstrates the effectiveness of the model in the missing boundary conditions scenario. In summary, the U-Net-ConvLSTM framework is capable of making precise spatiotemporal predictions in hydrodynamics, which may be considered a high-performance computational solution for predicting hydrodynamics at the basin scale.
... Three-dimensional hydrodynamic models based on the grid Euler's approach are available for modern computing systems [9,10]. The use of 3D models even for real shallow water bodies, such as the Azov Sea, shows the important role of accounting for vertical movements [11]. Engineering calculations of the impact of water flows on offshore and coastal structures require 3D modeling of a fluid with a free surface [10,12]. ...
... Linearization of the hydrodynamic equations of an incompressible stationary fluid of thickness H 0 gives the following dispersion equation for gravitational surface waves [27] where c (s) 0 = √ gH 0 . The limiting case kH 0 1 in (11) gives ω = √ kg. The frequency in the inverse long-wavelength limit (kH 0 1) is ...
Software has been developed for 3D modeling of fluid motion on an inhomogeneous terrain for problems of river hydrology. Parallelization of the Smoothed-particle hydrodynamics (SPH) method was performed using both OpenCL and CUDA for GPU computing systems. The efficiency of parallelization as a function of the number of SPH particles has been studied for various GPUs. We implemented an approach based on immobile SPH particles for setting boundary conditions on a geometrically complex surface. The digital elevation model defines such a boundary surface using standard 3D geometric solid modeling techniques. The dispersion properties of surface gravitational linear waves in the numerical model are in good agreement with the exact solution for a wide range of wavenumbers.
... Рис. 3. Линеаризация квадратичного члена трения [15] Обоснование линеаризации заключается в том факте, что она, как принято считать, не воспроизводит точную функцию косинуса до тех пор, пока сохраняется демпфирующий эффект трения. С этой целью энергия, которая теряется за цикл из-за трения, устанавливается равной для обоих случаев. ...
Introduction . Two-dimensional hydrodynamic models have proven their ability to adequately describe the processes of runoff and transportation in rivers, lakes, estuaries, deltas and seas. Practice shows that even where significant three-dimensional effects are expected, for example, with wind flows, a two-dimensional approach can work effectively. However, in some cases, the two-dimensional model does not accurately reflect the actual flow structures. For example, in shallow waters with complex bathymetry, heterogeneous terrain and dynamics can lead to a non-uniform velocity profile. The aim of the study is to develop a basis for determining in which cases a two-dimensional model averaged in depth is sufficient for modelling hydrodynamic processes in shallow waters like the Azov Sea, and in which cases it is advisable to use a three-dimensional model to obtain accurate results.
Materials and Methods. Local analytical solutions have been obtained for the propagation of the predominant singular progressive wave in a shallow, well-mixed reservoir. Advective terms and Coriolis terms are neglected, the vortex viscosity is assumed to be constant, and the lower friction term is linearized. Special attention is paid to the latter, since the characteristics of the models significantly depend on the method of determining the coefficients of lower friction. The analytical method developed in the study shows that certain combinations of higher flow velocities (u ≈˃ 1 m/s) and water depths (d ˃ 50 m) can cause significant differences between the results of the depth-averaged model and the model containing vertical information.
Results . The results obtained are verified by numerical simulation of stationary and non-stationary periodic flows in a schematized rectangular basin. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modelling averaged in depth. Both simulations show good compliance with analytical solutions.
Discussion and Conclusions. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects — nonlinearities caused by higher-order terms in the equations of motion, i.e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.
... The authors studied the mathematical aspects of the model of geochemical cycles and biological kinetics of a multispecies model of populations, considering the following factors: the movement of the water flow, the spatially uneven distribution of temperature and salinity, as well as the interaction of the main biogenic substances-compounds of nitrogen, phosphorus, and the main species of plankton populations, including their growth, reproduction, natural decrease in numbers, etc. The input data for the model of biogeochemical processes are of the form of the velocity vector of the aquatic environment, which is calculated on the basis of the model of hydrodynamics of coastal systems [30]. Combining these two models allows us to simulate the researched processes in a medium with a significant density gradient and a large difference in depths and considers the complex shape of the computational domain. ...
The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L2 as the time step τ tends to zero at the rate O(τ).
... The results of numerical calculations of the problem of transport of substances on the basis of the developed scheme are presented. The proposed threedimensional hydrodynamic model has been repeatedly used for diagnostic and predictive calculations of the water flow vector velocity in shallow water bodies, for example, in [11] for the Lagoon Etang de Berre (south of France) and the Azov Sea (south of Russia). ...
... To calculate the components of the velocity vector of the water medium, a threedimensional model of the hydrodynamics of shallow reservoirs was used [11,23]. ...
In recent years, the number of adverse and dangerous natural and anthropogenic phenomena has increased in coastal zones around the world. The development of mathematical modeling methods allows us to increase the accuracy of the study of hydrodynamic processes and the prediction of extreme events. This article discusses the application of the modified Upwind Leapfrog scheme to the numerical solution of hydrodynamics and convection–diffusion problems. To improve the accuracy of solving the tasks in the field of complex shapes, the method of filling cells is used. Numerical experiments have been carried out to simulate the flow of a viscous liquid and the transfer of substances using a linear combination of Upwind and Standard Leapfrog difference schemes. It is shown that the application of the methods proposed in the article allows us to reduce the approximation error in comparison with standard schemes in the case of large grid numbers of Péclet and to increase the smoothness of the solution accuracy at the boundary. The soil dumping and suspended matter propagation problems are solved using the developed schemes.
... The Azov Sea, a unique water body in the South of Russia, has been chosen as a modeling object, for which this model was verified [17,[27][28][29][30]. The choice of this water body for the study is not accidental. ...
... To approximate the convective terms in the diffusion-convection-reaction equations, improved Upwind Leapfrog schemes are used, which have better accuracy and a large margin of stability in comparison with those known for large values of the grid Péclet number. To discretize a continuous mathematical model of the dynamics of the most common in Azov Sea phytoplankton summer species (1)-(4), a linear combination of a central difference scheme and an Upwind Leapfrog scheme with weight parameters selected based on minimizing the approximation error were used [29]. The conservativeness of the proposed difference scheme at the discrete level is studied. ...
The article considers a three-dimensional mathematical model of population dynamics based on a system of non-stationary parabolic advection-diffusion-reaction equations with lower derivatives describing the advective motion of the aquatic environment and non-linear source functions. In contrast to the previous authors’ works devoted to the description of this model and its numerical implementation, this article presents the results of an analytical study of the initial-boundary value problem corresponding to this model. For these purposes, the original initial-boundary value problem is linearized on a single time grid—for all nonlinear sources, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial—final data at each step of the time grid. For this chain of linearized problems, the existence and uniqueness of the solution of the initial-boundary value problem for the system of partial differential equations in the Hilbert space were researched. Numerical experiments were performed for model problems of the main types of phytoplankton populations in coastal systems under the influence of dynamically changing biotic and abiotic factors, the results of which are consistent with real physical experiments. The developed model, including the proposed model of biological kinetics, allows for the study of the productive and destructive processes of the shallow water body biocenosis to assess the state of the processes of reproduction of valuable and commercial fish participating in the food chain with selected species of summer phytoplankton.
... To solve two-dimensional grid equations in a two-dimensional-one-dimensional scheme, the authors previously used the adaptive iterative alternating-triangular method developed by them for problems with a non-self-adjoint operator. The articles [17,29] are devoted to the study of its characteristics. ...
The initial boundary value problem for the 3D convection-diffusion equation corresponding to the mathematical model of suspended matter transport in coastal marine systems and extended shallow water bodies is considered. Convective and diffusive transport operators in horizontal and vertical directions for this type of problem have significantly different physical and spectral properties. In connection with the above, a two-dimensional–one-dimensional splitting scheme has been built—a three-dimensional analog of the Peaceman–Rachford alternating direction scheme, which is suitable for the operational suspension spread prediction in coastal systems. The paper has proved the theorem of stability solution with respect to the initial data and functions of the right side, in the case of time-independent operators in special energy norms determined by one of the splitting scheme operators. The accuracy has been investigated, which, as in the case of the Peaceman–Rachford scheme, with the special definition of boundary conditions on a fractional time step, is the value of the second order in dependency of time and spatial steps. The use of this approach makes it possible to obtain parallel algorithms for solving grid convection-diffusion equations which are economical in the sense of total time of problem solution on multiprocessor systems, which includes time for arithmetic operations realization and the one required to carry of information exchange between processors.
... For discretization, we use a linear combination of central and Upwind Leapfrog difference schemes [14], which will improve the approximation accuracy at large values of the grid Péclet number (Peh > 2).When constructing the difference equations, we will take into account the filling of the cells [15], which allows to reduce the approximation error at the boundary. A discrete analogue for equation ( Experiments have shown that the considered scenario is valid for the typical development of phytoplankton populations in the Azov Sea in summer, and allows us to analyze the self-purification process as a result of bacteria development involved in the decomposition of detritus and the reduction of sulfate to hydrogen sulfide as a result of anaerobic respiration of microorganisms, the influence of both biotic and external factors, including illumination, salinity and temperature, on the production and destruction processes of phytoplankton. ...
The paper covers the model of shallow water self-purification processes. The proposed mathematical model of biological kinetics is based on a system of non-stationary convection-diffusion-reaction equations with nonlinear terms, taking into account the water flow movement, gravitational sedimentation of impurities, microturbulent diffusion, and the detritus decomposition as a result of activity the aerobic and anaerobic bacteria. Discretization is performed on the basis of a linear combination of central and Upwind Leapfrog difference schemes, which makes it possible to increase the solution accuracy of biological kinetics problem at large values of the grid Péclet number (Peh > 2). To solve high-dimensional SLAEs, a modified alternating-triangular method was used.
... A uniform rectangular space-time grid was used. To solve the obtained SLAE, a modified alternating triangular method was used [12]. Results and discussion Numerical experiments were carried out to simulate biogeochemical processes in the Azov Sea in the summer. ...
In the article biogeochemical processes of the Azov Sea were researched. Mathematical non-stationary 3D model is proposed which describes the development dynamics of the two most common species of phytoplankton populations in the summer, the growth of which is limited by a single biogenic element, is proposed the linearization of continuous mathematical model on a uniform temporal grid is made. For a continuous model, a discrete analogue is constructed and an optimal method for grid equations solving is selected. To determine the boundary of the considered computational domain of a complex shape an image processing algorithm has been developed, implemented as a software module on Python, which makes it possible to obtain a dynamically changing contour of the Azov Sea from satellite images.
... Thus, the turbulent flow can be divided into an average (deterministic) u and ripple u components u u u . Turbulent flows, in which the averaged component does not depend on time, are called stationary [4]. [5]. ...
The work describes research of vertical turbulent exchange structure and parametrization for 3D shallow water hydrodynamics models. In this paper, the coefficients of horizontal turbulent exchange are calculated using a whole set of averaging periods of turbulent velocity pulsations. Using experimental data on the pulsations of the velocity components, the coefficient of vertical turbulent exchange was calculated on the basis of various approaches to its parameterization, based on the analysis of the obtained coefficient distributions, the most adequate parameterization of the coefficient was selected, which is used in the software package. The distribution of the vertical turbulent exchange coefficient obtained in a numerical experiment was compared with the results of full-scale measurements, and the calculation results obtained using the mathematical statistics apparatus were analysed.
