Keywords

1 Introduction

The current environmental condition of the coastal systems is largely determined by the diversity of incoming solid particles of both mineral and organic origin. The suspended matter particles are transformed and deposited and, as a consequence, the bottom sediment is formed under the influence of complex of intra-water processes. The coastal systems’ sediments are a complex heterogeneous physical and chemical system with the ongoing processes that are especially relevant to study now [1, 2]. It is relevant to assess the pollutant flows and determine the amount of sediments associated with the development of industrial and recreational activities in the coastal areas [3]. As a rule, this area is researched with the help of mathematical models aligned with the real processes and making it possible to predict the suspended matter distribution in the aquatic environment [4].

The complexity of the sediment transport processes in the aquatic environment of the coastal systems and the need to model them on detailed grids of \(10^6\)–\(10^9\) nodes requires the use of parallel computing systems (more than 100 teraflops) to do the operational forecasting of these processes. In this work, the authors proposed and investigated a discrete spatially two-dimensional model of sediment transport, produced by approximating the corresponding linearized continuous model and supplemented with the Navier–Stokes and continuity equations, and the aquatic environment condition equation. The model takes into account the following physical parameters and processes: soil porosity; the critical shear stress of the commencement of the sediment movement; turbulent exchange; dynamically changing bottom geometry and elevation function; wind currents; bottom friction [5,6,7,8]. To achieve a joint numerical solution of the sediment transport and wave hydrodynamics problems using a supercomputing system with distributed memory with a relatively small number of cores (up to 2048), using parallel algorithms is considered. The authors also investigated the issues of parallelizing the numerical solution processes on massively parallel systems that provide high efficiency algorithms for the systems containing many tens of thousands of cores. To solve the system of grid equations, an adaptive modified alternately triangular iterative method was used. The results of numerical experiments are presented.

2 Continuous 2D Model of Sediment Transport

Water flows carry a large amount of solid sediment particles, moving particles of clay, mud, gravel, pebbles, sand, loess, carbonate compounds, mineral oil emulsions, petroleum products and other components. The surfaces of the sediment particles are capable of absorbing various pollutants, including heavy metals and pesticides which make a negative impact on the ecological situation of the water body [9, 10]. Sediment can be carried by the flow in a suspended state (suspended sediment), and can be moved in the bottom layer of the stream by rolling, sliding, saltation (entrained sediment). Particles of the flow-carried sediment can transform from the suspended state to the entrained state and vice versa, entrained particles can stop moving, and motionless particles can begin moving. The nature of the movement of suspended and entrained sediments are determined by the flow velocity, depth, and other hydraulic elements of the water flow.

3 Mathematical Description of the Initial Boundary Problem of Sediment Transport

Let us consider the sediment transport equation [11, 12]:

$$\begin{aligned} \left( 1-\varepsilon \right) \frac{\partial H}{\partial t} =div\left( k\frac{\tau _{bc} }{\sin \varphi _{0} } gradH\right) -div\left( k\mathbf {\tau }_{b} \right) , \end{aligned}$$
(1)

where \(H=H\left( x,y,t\right) \) is the depth of the pond; \(\varepsilon \) is the porosity of bottom materials; \(\mathbf {\tau }_{b} \) is the tangential stress vector at the bottom of the reservoir; \(\tau _{bc} \) is the critical tangential stress value; \(\tau _{bc} =a\sin \varphi _{0} \), \(\varphi _{0} \) is the angle of the soil repose in the pond; \(k=k\left( H,x,y,t\right) \) is a nonlinear coefficient defined by the relation:

$$\begin{aligned} k=\frac{\mathrm{A}\tilde{\omega }\mathrm{d}}{\left( \left( \rho _{1} -\rho _{0} \right) gd\right) ^{\beta } } \left| \mathbf {\tau }_{b} -\frac{\tau _{bc} }{\sin \varphi _{0} } gradH\right| ^{\beta -1}, \end{aligned}$$
(2)

where \(\rho _{1},\rho _{0} \) are the particle densities of the bottom material and the aqueous medium, respectively; g an acceleration of gravity; \(\tilde{\omega }\) is wave frequency; \(\mathrm{A}\) and \(\beta \) are the dimensionless constants; d are the characteristic sizes of the soil particles.

We supplement Eq. (1) with the initial condition assuming that the function of the initial conditions belongs to the corresponding smoothness class:

$$\begin{aligned} \begin{gathered} H\left( x,y,0\right) =H_{0} \left( x,y\right) , H_{0} \left( x,y\right) \in C^{2} \left( D\right) \cap C\left( \overline{D}\right) , \\ grad_{\left( x,y\right) } H_{0} \in C\left( \overline{D}\right) ,\left( x,y\right) \in \overline{D}. \end{gathered} \end{aligned}$$
(3)

Let us formulate the region \(\bar{D}\) boundary conditions as:

$$\begin{aligned} \left. \left| \overrightarrow{\tau _{b} }\right| \, \right| _{y=0} =0, \end{aligned}$$
(4)
$$\begin{aligned} H\left( L_{x} ,y,t\right) =H_{2} \left( y,t\right) ,\, \, \, 0\le y\le L_{y}. \end{aligned}$$
(5)
$$\begin{aligned} H\left( 0,y,t\right) =H_{1} \left( y,t\right) ,\, \, \, 0\le y\le L_{y}, \end{aligned}$$
(6)
$$\begin{aligned} H\left( x,0,t\right) =H_{3} \left( x\right) ,\, \, \, 0\le x\le L_{x}. \end{aligned}$$
(7)
$$\begin{aligned} H\left( x,L_{y} ,t\right) =0,\, \, \, 0\le x\le L_{x}. \end{aligned}$$
(8)

In addition to the boundary conditions (5)–(8), assume that their smoothness conditions are satisfied, and continuous derivatives of the H function on the boundary of the region D exist:

$$\begin{aligned} grad_{\left( x,y\right) } H\in C\left( \overline{CY}_{T} \right) \cap C^{1} \left( CY_{T} \right) . \end{aligned}$$
(9)

The condition of non-degeneracy operator has the form:

$$\begin{aligned} k\ge k_{0} =const>0,\, \, \forall \left( x,y\right) \in \overline{D},\, \, \, 0<t\le T. \end{aligned}$$
(10)

The vector of tangential stress at the bottom is expressed using unit vectors of the coordinate system in a natural way:

$$\begin{aligned} \mathbf {\tau }_{b} =\mathbf {i}\tau _{bx} +\mathbf {j}\tau _{by}, \tau _{bx} =\tau _{bx} \mathrm{\; }\left( x,y,t\right) , \tau _{by} =\tau _{by} \mathrm{\; }\left( x,y,t\right) . \end{aligned}$$
(11)

4 Construction of the Linearized Initial-Boundary Value Problem of Sediment Transport

We construct linearized chain of sediment transport problems [13] on a uniform time grid \(\omega _{\tau } =\left\{ t_{n} =n\tau ,\, \, n=0,1,...,N,\, \, \, N\tau =T\right\} \), approximating with an error \(O\left( \tau \right) \) in Hilbert space \(L_{1} \left( D\times \left[ 0,T\right] \right) \) initial initial-boundary value problem (1)–(8).

Introduce the following notation:

$$\begin{aligned} k^{\left( n-1\right) } \equiv \frac{\mathrm{A}\tilde{\omega }\mathrm{d}}{\left( \left( \rho _{1} -\rho _{0} \right) gd\right) ^{\beta } } \left| \overrightarrow{\tau }_{b} -\frac{\tau _{bc} }{\sin \varphi _{0} } gradH^{\left( n-1\right) } \left( x,y,t_{n-1} \right) \right| ^{\beta -1}, \end{aligned}$$
(12)
$$\begin{aligned} n=1,2,...,N. \end{aligned}$$

Then Eq. (1) after linearization, will take the form:

$$\begin{aligned} \left( 1-\varepsilon \right) \frac{\partial H^{\left( n\right) } }{\partial t} =div\left( k^{\left( n-1\right) } \frac{\tau _{bc} }{\sin \varphi _{0} } gradH^{\left( n\right) } \right) -div\left( k^{\left( n-1\right) } \overrightarrow{\tau }_{b} \right) , \end{aligned}$$
(13)
$$\begin{aligned} \mathrm{t}_{n-1} <t\le \mathrm{t}_{n} ,\, \, n=1,...,N \end{aligned}$$

and add the initial conditions:

$$\begin{aligned} H^{\left( 1\right) } \left( x,y,t_{0} \right) =H_{0} \left( x,y\right) ,\, H^{\left( n\right) } \left( x,y,t_{n-1} \right) =H^{\left( n-1\right) } \left( x,y,t_{n-1} \right) , \end{aligned}$$
(14)
$$\begin{aligned} \left( x,y\right) \in \overline{D},\, \, n=2,...,N. \end{aligned}$$

\(div\, \left( k^{\left( n-1\right) } \overrightarrow{\tau }_{b} \right) \) is a known function of the right-hand side with such linearization; boundary conditions (4)–(8) are assumed to be satisfied for all time intervals \(\mathrm{t}_{n-1} <t\le \mathrm{t}_{n}, n=1,2,...,N\).

The coefficients \(k^{\left( n-1\right) }, n=1,2,...,N\) depend on spatial variables x, y and time variable \(t_{n-1} \mathrm{,} n=1,2,...,N\), determined by the choice of grid step \(\tau \), i.e. \(k^{\left( n-1\right) } =k^{\left( n-1\right) } \left( x,y,t_{n-1} \right) \mathrm{,}\, \, \, n=1,2,...,N\).

In [14], the conditions for the existence and uniqueness of the sediment transport problem under conditions of smoothness of the solution function were studied

$$\begin{aligned} H\left( x,y,t\right) \in C^{2} \left( CY_{T} \right) \cap C\left( \overline{CY}_{T} \right) ,\, \, \, gradH\in C\left( \overline{CY}_{T} \right) \end{aligned}$$

and the necessary smoothness of the region boundary, as well as the a priori estimate of the solution in the norm of the space \(L_{1} \) depending on the integral estimates of the right-hand side, boundary conditions and the norm of the initial condition [18, 19] were considered. The results of the convergence of the linearized problem solution to the solution of the initial nonlinear initial-boundary problem of sediment transport in the norm of the space were presented \(L_{1} \) with the time grid step of the linearization tending to zero [15].

5 Spatially Heterogeneous 3D Model of Hydrodynamics

The input data for the sediment transport model is the velocity vector of the aquatic environment. To calculate tangential stresses in the sediment transport model, the velocity vector data for the water medium at the reservoir bottom is needed. The initial hydrodynamic model equations are [16, 17]:

  • equation of motion (Navier-Stokes):

    $$\begin{aligned} \begin{array}{c} {u'_{t} +uu'_{x} +vu'_{y} +wu'_{z} =-\frac{1}{\rho } P'_{x} +\left( \mu u'_{x} \right) _{x} ^{{'} } +\left( \mu u'_{y} \right) _{y} ^{{'} } +\left( \nu u'_{z} \right) _{z} ^{{'} } ,} \\ {v'_{t} +uv'_{x} +vv'_{y} +wv'_{z} =-\frac{1}{\rho } P'_{y} +\left( \mu v'_{x} \right) _{x} ^{{'} } +\left( \mu v'_{y} \right) _{y} ^{{'} } +\left( \nu v'_{z} \right) _{z} ^{{'} } ,} \\ {w'_{t} +uw'_{x} +vw'_{y} +ww'_{z} =-\frac{1}{\rho } P'_{z} +\left( \mu w'_{x} \right) _{x} ^{{'} } +\left( \mu w'_{y} \right) _{y} ^{{'} } +\left( \nu w'_{z} \right) _{z} ^{{'} } +g;} \end{array} \end{aligned}$$
    (15)

  • continuity equation for variable density cases:

    $$\begin{aligned} \rho _{t}^{{'} } +\left( \rho u\right) _{x}^{{'} } +\left( \rho v\right) _{y}^{{'} } +\left( \rho w\right) _{z}^{{'} } =0, \end{aligned}$$
    (16)

where \(V=\left\{ u,v,w\right\} \) is the velocity vector of the water current in a shallow water body; \(\rho \) is the aquatic environment density; P is the hydrodynamic pressure; g is the gravitational acceleration; ;  \(\mu \), \(\nu \) are coefficients of turbulent exchange in the horizontal and vertical directions; n is the normal vector to the surface describing the boundary of the computational domain.

Add boundary conditions to system (1)–(2):

  • entrance (left border): \(\mathbf{V}=\mathbf{V}_{0} ,\, \, P'_{n} =0,\)

  • bottom border: \(\rho \mu \left( \mathbf{V}_{\mathbf{\tau }} \right) ^{{'} } _{n} =-{\varvec{\tau }},\, \, \mathbf{V}_{n} =0,\, \, P'_{n} =0,\)

  • lateral border: \(\left( \mathbf{V}_{\mathbf{\tau }} \right) ^{{'} } _{n} =0,\, \, \mathbf{V}_{n} =0,\, \, P'_{n} =0,\)

  • upper border: \(\rho \mu \left( \mathbf{V}_{\mathbf{\tau }} \right) ^{{'} } _{n} =-{\varvec{\tau }},\, \, \, w=-\omega -\, P'_{t} /\rho g,\, \, P'_{n} =0,\)

  • surface of the structure: \(\rho \mu \left( \mathbf{V}_{\mathbf{\tau }} \right) ^{{'} } _{n} =-{\varvec{\tau }},\, w=0,\, \, P'_{n} =0,\)

where \(\omega \) is the liquid evaporation intensity, \( \mathbf {V}_{n}, \mathbf {V}_{\mathbf{\tau }} \) are the normal and tangential components of the velocity vector, \({\varvec{\tau }}=\left\{ \tau _{x},\tau _{y},\tau _{z} \right\} \) is the tangential stress vector.

Let \({\varvec{\tau }}=\rho _{a} Cd_{s} \left| w\right| w\), where w is the wind velocity relative to water, \(\rho _{a }\) is the atmosphere density, \(Cd_{s} \) = 0.0026.

Let us set the tangential stress vector for the bottom taking into account the movement of water as follows: \({\varvec{\tau }}=\rho Cd_{b} \left| \mathbf{V}\right| \mathbf{V}\), \(Cd_{b} =gk^{2} /h^{1/3} \), where \(k=0,04\) is the group roughness coefficient in the Manning formula, considered in the range of 0,025–0,2. Note that the magnitude of the coefficients \(Cd_{s}\), \(Cd_{b}\) is influenced by many parameters, including wind speed, stratification, age of sea waves, wind direction, sea depth, roughness of the bottom surface, the shape of the bottom, siltation and erosion processes, the presence of obstacles, etc. factors. In the coastal part of the reservoir, as a rule, an increase in the ratio \(Cd_{s}\) compared with values in the deep sea, due to a decrease of the phase velocity of the waves, increasing their steepness, fast changing wave field, a growing number of collapses, bottom topography, nature of the shoreline. It is believed that with a decrease in depth, the value of the coefficient \(Cd_{b}\) also increases. Taking into account the influence of the above factors on the nature of the coefficients \(Cd_{s}\), \(Cd_{b}\) is quite difficult and, therefore, they are the result of processing numerous experimental data obtained experimentally.

6 Discrete Model

Let construct a finite-difference scheme approximating problem (13), (14), (4)–(8). Cover the area D uniform rectangular calculation grid \(\omega =\omega _{x} \times \omega _{y} \), assuming that the time grid \(\omega _{\tau } \) previously defined.

$$\begin{aligned} \begin{array}{c} {\omega _{x} =\left\{ x_{i} =ih_{x} ,\, \, 0\le i\le N_{x} -1,\, \, l_{x} =h_{x} \left( N_{x} -1\right) \right\} ,} \\ {\omega _{y} =\left\{ y_{j} =jh_{y} ,\, \, 0\le j\le N_{y} -1,\, \, l_{y} =h_{y} \left( N_{y} -1\right) \right\} ,} \end{array} \end{aligned}$$

where n, i, j are indices of grid nodes constructed on a temporary Ot and spatial Ox, Oy directions, respectively, \(\tau , h_{x}, h_{y} \) are grid steps in temporal and spatial directions, respectively, \(N_{t},\; N_{x},\; N_{y} \) are the number of nodes in the temporal and spatial directions, respectively.

The balance method was used to obtain a difference scheme. We integrate both sides of Eq. (13) over the region \(D_{txy} \):

$$\begin{aligned} D_{txy} \in \left\{ t\in \left[ t_{n}, t_{n+1} \right] ,\, x\in \left[ x_{i-1/2} ,x_{i+1/2} \right] ,\, y\in \left[ y_{j-1/2} ,y_{j+1/2} \right] \right\} , \end{aligned}$$

as a result, we obtain the following equality:

$$\begin{aligned} \begin{array}{c}\int {\int {\int _{D_{txy}}}}\left( 1-\varepsilon \right) H_{t}^{{\left( n\right) }^{{'}}} dtdxdy+\int {\int {\int _{D_{txy} }}}\left( k^{\left( n-1\right) } \tau _{b,x} \right) _{x}^{'} dtdxdy \\ +\int {\int {\int _{D_{txy} }}}\left( k^{\left( n-1\right) } \tau _{b,y} \right) _{y}^{'} dtdxdy \\ =\int {\int {\int _{D_{txy} }}}\left( k^{\left( n-1\right) } \frac{\tau _{bc} }{\sin \varphi _{0} } H_{x}^{{\left( n\right) }^{{'}} } \right) _{x}^{'} dtdxdy \\ +\int {\int {\int _{D_{txy} }}}\left( k^{\left( n-1\right) } \frac{\tau _{bc} }{\sin \varphi _{0} } H_{y}^{{\left( n\right) }^{{'} }} \right) _{y}^{'} dtdxdy. \end{array} \end{aligned}$$
(17)

In equality (17), we calculate the approximate integrals from the rectangle formulas, divide the resulting equation by the product of multipliers \(\tau , h_{x}, h_{y} \), and, replacing the approximate equality with the exact one, we obtain a difference scheme approximating the linearized continuous problem:

$$\begin{aligned} \begin{array}{c} {\left( 1-\varepsilon \right) \frac{H_{i,j}^{\left( n+1\right) } -H_{i,j}^{\left( n\right) } }{\tau } } \\ {+\,\frac{k_{i+1/2,j}^{\left( n\right) } \left( \tau _{b,x} \right) _{i+1/2,j}^{\left( n\right) } -k_{i-1/2,j}^{\left( n\right) } \left( \tau _{b,x} \right) _{i-1/2,j}^{\left( n\right) } }{h_{x} } +\frac{k_{i,j+1/2}^{\left( n\right) } \left( \tau _{b,y} \right) _{i,j+1/2}^{\left( n\right) } -k_{i,j-1/2}^{\left( n\right) } \left( \tau _{b,y} \right) _{i,j-1/2}^{\left( n\right) } }{h_{y} } } \\ {=\frac{\tau _{bc} }{\sin \varphi _{0} } \left( k_{i+1/2,j}^{\left( n\right) } \frac{H_{i+1,j}^{\left( n+\sigma \right) } -H_{i,j}^{\left( n+\sigma \right) } }{h_{x}^{2} } -k_{i-1/2,j}^{n} \frac{H_{i,j}^{\left( n+\sigma \right) } -H_{i-1,j}^{\left( n+\sigma \right) } }{h_{x}^{2} } \right) } \\ {+\frac{\tau _{bc} }{\sin \varphi _{0} } \left( k_{i,j+1/2}^{\left( n\right) } \frac{H_{i,j+1}^{\left( n+\sigma \right) } -H_{i,j}^{\left( n+\sigma \right) } }{h_{y}^{2} } -k_{i,j-1/2}^{n} \frac{H_{i,j}^{\left( n+\sigma \right) } -H_{i,j-1}^{\left( n+\sigma \right) } }{h_{y}^{2} } \right) ,} \end{array} \end{aligned}$$
(18)

where

$$\begin{aligned} \left( \tau _{b,x} \right) _{i+1/2,j}^{\left( n\right) } =\frac{\left( \tau _{b,x} \right) _{i+1,j}^{\left( n\right) } +\left( \tau _{b,x} \right) _{i,j}^{\left( n\right) } }{2} ,\left( \tau _{b,y} \right) _{i,j+1/2}^{\left( n\right) } =\frac{\left( \tau _{b,y} \right) _{i,j+1}^{\left( n\right) } +\left( \tau _{b,y} \right) _{i,j}^{\left( n\right) } }{2}, \end{aligned}$$
$$k_{i+1/2,j}^{\left( n\right) } \!=\!\frac{A\varpi d\left| \left( \overrightarrow{\tau }_{b} \right) _{i+1/2,j}^{\left( n\right) } \!-\!\frac{\tau _{bc} }{\sin \varphi _{0} } \left( gradH\right) _{i+1/2,j}^{\left( n\right) } \right| }{\left( \left( \rho _{1} -\rho _{0} \right) gd\right) ^{\beta } } h\left( \left| \left( \overrightarrow{\tau }_{b} \right) _{i+1/2,j}^{\left( n\right) } \!-\!\frac{\tau _{bc} }{\sin \varphi _{0} } \left( gradH\right) _{i+1/2,j}^{\left( n\right) } \right| -\tau _{bc} \right) \!.$$

The value \(\left. gradH\right| _{\left( x_{i+1/2} ,y_{j} \right) } \) is written as

$$\begin{aligned} \left( gradH\right) _{i+1/2,j} =\frac{H_{i+1,j} -H_{i,j} }{h_{x} } \overrightarrow{i}+\frac{H_{i+1/2,j+1} -H_{i+1/2,j-1} }{2h_{y} } \overrightarrow{j}. \end{aligned}$$

Similarly, you can get the following approximation:

$$\begin{aligned} \left( gradH\right) _{i,j+1/2} =\frac{H_{i+1,j+1/2} -H_{i-1,j+1/2} }{2h_{x} } \bar{i}+\frac{H_{i,j+1} -H_{i,j} }{h_{y} } \bar{j}. \end{aligned}$$

Difference schemes are investigated for stability using the grid maximum principle. The constructed difference schemes are stable under the following restriction on the time step:

$$\begin{aligned} \tau <\frac{\sin \varphi _{0} \left( 1-\varepsilon \right) }{\tau _{bc} \left( 1-\sigma \right) \mathop {\max }\limits _{0\le m\le N-1} \left\{ k\left( t_{m} \right) \right\} \left( \frac{2}{h_{x}^{2} } +\frac{2}{h_{y}^{2} } \right) }. \end{aligned}$$

The upper bound for the grid function is the solution of the difference problem in the grid norm \(C_{h,\tau } \) formulated as:

$$\begin{aligned} \left\| H^{n} \right\| _{c_{h,\tau } } \le \left\| H^{0} \right\| _{c_{h} } +\max \left( \left\| H_{1} \right\| _{c_{h,\tau } },\, \left\| H_{2} \right\| _{c_{h,\tau } } ,\, \left\| H_{3} \right\| _{c_{h,\tau } } \right) \end{aligned}$$
$$\begin{aligned} +\frac{\tau }{1-\varepsilon } \sum _{m=0}^{n}\left\| \left( k\left( t_{m} \right) \tau _{b,x} \right) _{\mathop {x}\limits ^{0} }^{m} +\left( k\left( t_{m} \right) \tau _{b,y} \right) _{\mathop {y}\limits ^{0} }^{m} \right\| _{c_{h,\tau } } , \end{aligned}$$

which guarantees the stability of the constructed difference scheme with respect to the function of the right-hand side, the boundary and initial conditions.

The approximation error of the discrete sediment transport model was found that an order of magnitude \(O\left( \tau +h_{x}^{2} +h_{y}^{2} \right) \).

7 Parallel Algorithm Description

A software package in C++ is designed to build turbulent flows of an incompressible velocity field of the aquatic environment on high-resolution grids for predicting sediment transport and possible scenarios of changing the geometry of the bottom region of shallow water bodies. Parallel algorithms implemented in the software package for solving systems of grid equations arising during the discretization of model problems were developed using MPI technology.

To solve this problem, the adaptive modified alternating-triangular method of minimum corrections was used. In parallel implementation, decomposition methods of grid domains were applied to computationally time-consuming diffusion-convection problems with respect to the architecture and parameters of a multiprocessor computing system. The calculated two-dimensional region was decomposed with two spatial variables x, y. The peak performance of the multiprocessor computing system is 18.8 teraflops. As computing nodes, 128 HP ProLiant BL685c homogeneous 16-core Blade servers of the same type were used, each being equipped with four 4-core AMD Opteron 8356 2.3 GHz processors and 32 GB RAM.

Figure 1 shows the dependence of acceleration on the number of processors needed to solve the model problem on various grids. The numbering of the graphs corresponds to the following dimensions of the calculation grids: 1–100 \(\times \) 100, 2–200 \(\times \) 200, 3–500 \(\times \) 500, 4–1000 \(\times \) 1000, 5–2000 \(\times \) 2000, 6–5000 \(\times \) 5000.

Fig. 1.
figure 1

Graph of the acceleration of the parallel algorithm (a – based on an explicit scheme, b – based on an implicit scheme

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Figure 1a shows that when using a parallel algorithm based on an explicit scheme, the maximum acceleration was achieved on 200 \(\times \) 200 grids. In this case, a superlinear acceleration arises due to the fact that with an increase in the number of computers, the total amount of their RAM and cache grows. Therefore, most of the task data is located in RAM and, moreover, is “placed” in the cache. In Fig. 1b shows a graph of the acceleration of the parallel algorithm based on the implicit scheme using the parallel version of the alternating triangular method of minimal corrections. The constructed graphs show that for each of the computational grids, the acceleration assumes the greatest value at a certain value of calculators, and with a further increase in the number of computing cores, the acceleration only decreases. This is due to a significant increase in time spent on data exchanges between computers. Parallel algorithms based on an explicit scheme are better parallelized than the methods of a parallel algorithm based on an implicit scheme using a parallel version of the alternating-triangular method of minimal corrections. Despite this, a parallel version of the modified alternating triangular method of minimal corrections is preferable for ill-conditioned problems, since it requires a significantly smaller number of iterations when solving the problem of transport of bottom materials.

Figure 2.A presented the graph of acceleration versus the number of processors needed to solve the problems of hydrodynamics is presented. To calculate the velocity vector of the aqueous medium, computational grids of size 50 \(\times \) 250 \(\times \) 40.

Fig. 2.
figure 2

Acceleration schedule of the hydrodynamic problems’ parallel solution algorithm

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The maximum acceleration of 43.668 was achieved on 128 cores, each running an MPI process. On 256 computers, a drop in acceleration was observed.

8 Numerical Experiments for Modeling Sediment Transport and Bottom Topography Dynamics

After the software package had been developed, a series of numerical experiments was performed to simulate the dynamically changing bottom topography of various configurations. In the model problems, the subsequent movement of the aquatic environment was calculated with respect to the irregularities of the bottom surface (hilly terrain, boulders, terraces, underwater valleys, underwater breakwaters, dams, etc.).

Fig. 3.
figure 3

The geometry of the computational domain at the initial moment

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The paper presents the results of modeling the bottom change dynamics for the case of presence of pointed structures, such as discontinuous dams, acting as obstacles on the bottom surface. Retaining the sediment, dams do not only stop the movement of the material carried by the waves along the shore, but also contribute to its deposition.

The simulation section under consideration has dimensions of 55 m by 55 m horizontally and 2 m vertically (in depth), with the peak point rising 1 m above sea level. Suppose that the liquid is at rest at the initial time. The computational grid size is 110 by 110, the step in spatial variables is 0.05 m, the time step is 0.01 s, the wind is directed from left to right at the speed of 5 m/s.

Fig. 4.
figure 4

The geometry of the computational domain after 5 min from the start of the simulation.

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Figure 3 represents the initial position of the contour lines of the depth function (top) and bottom topography (bottom) featuring three pointed structures with an uneven surface. Structures of this kind can be, for example, moles, boulders, dumps, breakwaters. Fluctuations in the isolines of the depth function are observed in the central part of the computational domain. Being below the level of 0.2 m from the free surface, these structures are located at a distance of about 10 m from each other.

Sediment transport process modeling showed that smoothing of surface irregularities, sediment formation, and the decreasing repose angle of the coastal zone bottom that occur over time cause a gradual shallowing of the considered reservoir zone. For this reason, after 5 min, in the center of the computational domain the depth function contours took on a more tortuous shape, the pointed structures deformed and took the form of gentle slides, and the depth of the coastal zone reduced (Fig. 4).

Fig. 5.
figure 5

The computational domain geometry 15 min after the modeling start (a - isolines of the depth function, b - bottom relief)

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According to Fig. 5, the result of the described processes gets more visible at the process simulation time of 15 min. The depth function contours acquire a soft wave-like shape in the center of the computational domain, including the region of peak values. Along the coast, there is a zone of sediment formation and a decrease in the level of depth. The height of the slides decreases, and the slides themselves acquire a more smoothed appearance.

9 Conclusion

The experiment results provide the materials to analyze the dynamics of changes in the bottom surface shape, the function the aquatic environment elevation, the formation of ridges, sediments and the wave form of the functions of the bottom and the water surface. The proposed mathematical model and the developed software package can be used to predict the dynamics of the bottom surface behavior, the shape of marine braids and ridges, their growth and transformation. Therefore, the proposed mathematical models for sediment transport have been verified by the numerical experiment results.