The construction and investigation of parallel algorithms for the numerical realization of 3D models of transport and deposition of suspended matter and 2D models of bottom sediment transport in coastal marine systems have already developed. 3D model has been presented for numerical modeling of gravitational waves near the shoreline in this work. It consists of Navier-Stokes equation system, which includes three momentum equations and mass conservation law equation in region with dynamically varying boundaries. It has been shown that presented 3D model gives more realistic description of physical wave process near the coastal line. The practical signi cance of constructed 3D model, numerical algorithm and the complex of programs consist in the possibility of its application for the study of hydrophysical processes in coastal water systems, as well as in assessing of the hydrodynamic e ect on shore protection constructions and coastal structures in the presence of gravitational surface waves.
The study of the hydrodynamic processes of the coastal waters connected with the investigation of the in uence of wave processes generated in the open sea or in the coastal zone of the reservoir. The movement of waves can lead to negative results a ecting the operation of the coastal zone: to the transformation of the bottom surface resulting from the rise of bottom sediments, to abrasion, the process of destruction by the waves and surf of the banks of various water systems [Suk05]. The result of the interaction of waves with the bottom surface and the coastal slope is refraction, di raction and changes in wave structure. The most signi cant factors are uctuations in water surface level, wind phenomena, currents, transport of bottom materials and deformation of the coastal slope.
Characteristic feature of the coastal waters is the signi cant in uence of the bottom surface on the wave processes, which makes it di cult to study tidal phenomena in the coastal regions of the seas and river mouths [Gus02, Suk05']. The in uence of wave processes on the coastal zone can be ambivalent [Gus04]: wave processes can have a signi cant e ect on the accumulation and abrasion of the coastal zone of the reservoir and directly on coastal structures.
To simulate the hydrodynamic processes, the problem of practical application of computationally e ective methods is actualized, which makes it possible to obtain a fairly accurate approximate numerical solution [Ale13]. There is a need of constructing a set of interrelated models of three-dimensional wave processes intended for modeling wave processes. 2
The initial equations of hydrodynamics of shallow water bodies are [Suk05, Gus02, Suk05']: - the equation of motion (Navier-Stokes): u0t + uu0x + vu0y + wu0z = vt0 + uvx0 + vvy0 + wvz0 = wt0 + uwx0 + vwy0 + wwz0 = 1 p0x + ( u0x)0x + 1 p0y + ( vx0)0x + 1 p0z + ( wx0)0x + u0y 0y + ( u0z)0z ; vy0 0y + ( vz0)0z ; wy0 0y + ( wz0)0z + g; - the equation of continuity in the case of variable density:
0t + ( u)0x + ( v)0y + ( w)0z = 0;
Where V = fu; v; wg are the components of the velocity vector, p is the pressure, is the density, , are the horizontal and vertical components of the coe cient of turbulent exchange, g is the acceleration of gravity.
The system of equations (1)-(2) is considered under the following boundary conditions: - at the entrance - the lateral border (shore and bottom) u(x; y; z; t) = u(t); v(x; y; z; t) = v(t); p0n(x; y; z; t) = 0; Vn0 (x; y; z; t) = 0; = (u0)n(x; y; z; t) = x(t); (v0)n(x; y; z; t) = y(t); Vn(x; y; z; t) = 0; p0n(x; y; z; t) = 0; (1) (2) (3) (u0)n(x; y; z; t) = x(t); - the upper limit (v0)n(x; y; z; t) = y(t);
w(x; y; t) = ! p0t= g; p0n(x; y; t) = 0 where ! is the evaporation rate of the liquid, x; y are the components of the tangential stress [Gus04, Ale13]. The components of the tangential stress for the free surface: x = aCp (jw~ j) wx jw~ j, y = aCp (jw~ j) wy jw~ j, where w~ is the vector of the wind speed relative to the water, a is the density of the atmosphere, Cp (x) is the dimensionless coe cient. where U ; V are the time-averaged pulsations of the horizontal velocity components, is the characteristic scale of the grid, and Cs is the dimensionless empirical constant whose value is usually determined on the basis of calculating the decay process of homogeneous isotropic turbulence. 3
The computational domain inscribed in a parallelepiped. For the numerical realization of the discrete mathematical model of the hydrodynamic problem posed, a uniform grid is introduced: wh = ftn = n ; xi = ihx; yj = jhy; zk = khz; n = 0::Nt; i = 0::Nx; j = 0::Ny; k = 0::Nz;
Nt = T; Nxhx = lx; Nyhy = ly; Nzhz = lzg ; where is the time step, hx, hy hz are steps in space, Nt is the number of time layers, T is the upper bound on the time coordinate, Nx, Ny Nz is the number of nodes by spatial coordinates, lx, ly lz are boundaries along the parallelepiped in the direction of the axes Ox, Oy and Oz accordingly.
To solve the hydrodynamic problem, we used the method of correction to pressure. The variant of this method in the case of a variable density will take the form [Suk11, Bel75]: u~ v~ u v w~ w + uu0x + vu0y + wu0z = + uv0x + vv0y + wv0z = + uw0x + vw0y + ww0z =
0 u0x x +
0 v0x x +
0 w0x x +
0 u0y y +
0 v0y y +
0 w0y y +
0 u0z z ;
0 v0z z ;
0 w0z z + g; p^0x0x + p^0y0y + p^0z0z = u^ u~ = 1^p^0x;
^ v^ v~ 2 + = (^u~)0x + (^v~)0y + (^w~)0z
; 1^p^0y; w^ w~ = 1^p^0z; (4) (5) (6) where V = fu; v; wg are the components of the velocity vector, fu^; v^; w^g,fu~; v~; w~g are the components of the velocity vector elds on the new and intermediate time layers, respectively, u = (u~ + u) =2, ^ and is the distribution of the density of the aqueous medium on the new and previous time layers, respectively. In the construction of discrete mathematical models of hydrodynamics, the fullness of the control cells was taken into account, which makes it possible to increase the real accuracy of the solution in the case of a complex geometry of the investigated region by improving the approximation of the boundary.
Through oi; j;kmarked fullness of the cell (i; j; k) [Suk14]. The degree of fullness of the cell is determined by the pressure of the liquid column inside this cell. If the average pressure at the nodes that belong to the vertices of the cell in question is greater than the pressure of the liquid column inside the cell, then the cell is considered to be full (oi; j;k = 1). In the general case, the fullness of the cells can be calculated by the following formula [Suk11]: oi;j;k = pi;j;k + pi 1;j;k + pi;j 1;k + pi 1;j 1;k ; 4 ghz where p is the pressure.
We introduce the coe cients q0, q1, q2, q3, q4, q5, q6, describing the fullness of regions located in the vicinity of the cell (control areas). The value characterizes the fullness of the region D0: fx 2 (xi 1; xi+1), y 2 (yj 1; yj+1), z 2 (zk 1; zk+1)g, q1 { D1: fx 2 (xi; xi+1), y 2 (yj 1; yj+1), z 2 (zk 1; zk+1)g, q2 { D2: fx 2 (xi 1; xi), y 2 (yj 1; yj+1), z 2 (zk 1; zk+1)g, q3 { D3: fx 2 (xi 1; xi+1), y 2 (yj; yj+1), z 2 (zk 1; zk+1)g, q4 { D4: fx 2 (xi 1; xi+1), y 2 (yj 1; yj), z 2 (zk 1; zk+1)g, q5 { D5: fx 2 (xi 1; xi+1), y 2 (yj 1; yj+1), z 2 (zk; zk+1)g, q6 { D6: fx 2 (xi 1; xi+1), y 2 (yj 1; yj+1), z 2 (zk 1; zk)g. The lled parts of the regions Dm will be called qm can be calculated from the formulas:
m, where m = 0::6. In accordance with this, the coe cients (qm)i; j;k = S m ; (q0)i; j;k =
SDm h2x h2x hx
Similarly, approximations for the remaining coordinate directions will be recorded. The error in approximating the mathematical model is equal to O + khk2 ; where khk = qh2x + h2y + hz2. The conservation of the ow at the discrete level of the developed hydrodynamic model is proved, as well as the absence of non-conservative dissipative terms obtained as a result of discretization of the system of equations. A su cient condition for the stability and monotony of the developed model is determined on the basis of the maximum principle [Suk11], with constraints on the step with respect to the spatial coordinates: hx < j2 =uj ; hy < j2 =vj ; hz < j2 =wj or Re 2N; where Re = jV j l= are Reynolds numbers, l is the characteristic size of the region N = max fNx; Ny; Nzg.
Discrete analogs of the system of equations (5) are solved by an adaptive modi ed alternating-triangular method of variational type. 4
The resulting grid equations can be written in the matrix form [Suk12]: where A { a linear, positive de nite operator (A > 0). To nd the solution of problem (7) we will use an implicit iterative process
In equation (8), m is the iteration number, > 0 is an iteration parameter, and B is an invertible operator, called the preconditioner or stabilizer. The inversion of the operator B in (8) must be substantially simpler than the direct inversion of the original operator A in (7). In constructing B, we proceed from the additive representation of the operator A0 is the symmetric part of the operator A:
Ax = f; B xm+1
xm m+1
+ Axm = f: A0 = R1 + R2; R1 = R2; where A = A0 + A1; A0 = A0; A1 = A1.
The preconditioner operator will be written in the following form:
B = (D + !R1)D 1(D + !R2); D = D > 0; ! > 0; where D ia an operator.
Relations (9)-(
The algorithm of the adaptive modi ed alternating-triangular method of minimum corrections for calculating the grid equations with a nonselfadjoint operator has the form:
B(!m)wm = rm; rm = Axm f; !~m = s
(Dwm; wm) (D 1R2wm; R2wm) ; where rm is the discrepancy vector, wm is the correction vector, the diagonal part of the operator D is used as the operator. 5
Consider the parallel algorithm for calculating the correction vector [Suk12]:
(D + !mR1)D 1(D + !mR2)wm = rm; where R1 is the lower-triangular matrix, and R2 is the upper-triangular matrix. To this end, we solve successively the systems:
(D + !mR1)ym = rm; (D + !mR2)wm = Dym:
The constructed complex of programs allows you to set the shape and intensity of the source of oscillations, as well as the geometry of the surface object. Fig. 3 (a) shows the results of numerical experiments on modeling the propagation of wave hydrodynamic processes in the ow of an aquatic environment around a surface body, taking into account the geometries of the bottom of an object located in a liquid and the bottom of a reservoir. As an example of the practical use of a problem-oriented program complex, the problem of calculating the hydrodynamic e ect of support waves on structures is solved. Surface dimensions: 5 m wide, 10 m long, immersion depth 20 cm. The structure is installed at the bottom of the reservoir with the help of 6 supports. The selected section of the simulation has dimensions of 50x50 m and a depth of 1 m. The source of disturbances is set at some distance from the surface object. The boundaries of the computational domain are located so that the wave re ected from them does not change the parameters of the hydrodynamic force action on the surface structures. At the initial moment of time, the liquid is at rest. It is required to determine the subsequent movement of the aquatic environment when there is a surface object on the surface and hydrodynamic force loads on the structure supports. To solve this problem, a grid of 100x100 sizes was used, the time step is 0.01 seconds. surface body, there is a re ection of wave oscillations from a stationary object, which in turn leads to a change in the wave pro le. The source of oscillations is distributed along the left border and has a sinusoidal shape. The results of numerical experiments on modeling the propagation of hydrodynamic wave processes and provide an opportunity to assess the impact of waves on structures that have a support at the bottom of the reservoir. Fig. 4 shows the force hydrodynamic e ect on the supports of the surface structure installed at the bottom of the reservoir. The ordinate axis shows the power loads in tons, the abscissa axis indicates the time counted from the onset of oscillations in seconds.
On the basis of full-scale data, a three-dimensional model of wave hydrodynamic processes has been developed that describes the motion of an aquatic environment taking into account the wave's output to the shore. A modern software package adapted for simulation of hydrodynamic wave processes is developed, the eld of application of which is the construction of the velocity and pressure eld of the aquatic environment, and the evaluation of the hydrodynamic impact on the shore in the presence of surface waves. Based on the developed complex of programs, numerical simulation of hydrodynamic wave processes in the coastal zone of a shallow water body was carried out.
The practical signi cance of numerical algorithms and the complex of programs that realize them consists in the possibility of their application in the study of hydrophysical processes in coastal water systems, as well as in the construction of the velocity and pressure eld of the aquatic environment, and the evaluation of the hydrodynamic impact on the shore in the presence of surface waves. The constructed program complex allows you to specify the shape and intensity of the oscillation source, as well as the geometry of the bottom of the reservoir. Fig. 6 shows the results of numerical experiments on the simulation of the propagation of wave hydrodynamic processes when the wave leaves the shore, taking into account the geometries of the bottom of the object located in the liquid and bottom of the reservoir.
As an example of the practical use of a problem-oriented program complex, the problem of calculating the velocity and pressure elds is solved. The selected modeling site measures 50x50 m and a depth of 2 m, the peak point rises above sea level by 2 m. The disturbance source is given at some distance from the shore line. At the initial time, the liquid is at rest. To solve this problem, a grid of 100x100x40 dimensions was used, the time step is 0.01 seconds.
The work is devoted to the development of a model of three-dimensional wave processes, designed to simulate wave processes taking into account wave propagation towards the shore. A full-scale experiment was conducted to measure various parameters of wave propagation in shallow water. On the basis of experimental data, the values of the spectrum of the function of elevating the water level are obtained. The description of the developed program complex is given. The complex of programs constructed allows one to specify the shape and intensity of the source of oscillations, and also takes into account the ooding and drainage of coastal areas. On the basis of the developed software package, further studies of the calculation of the wave force e ects on surface objects and objects of coastal infrastructure and bottom surface geometry are possible. 7.0.1
Acknowledgements This work was supported by the Russian Foundation for Basic Research (project code 19-01-00701).