Additive two-dimensional splitting schemes for solving 3D suspension transport problems on optimal boundary-adaptive grids with uniform spacing’s in the vertical direction

The article considers3D matter transport model in dissolved and suspended forms (impurities) in coastal marine systems. The initial boundary value problem numerical solution is carried out on the basis of local2D splitting schemes. In this case, special attention is paid to the hig-hquality computational grid construction. As a rule, in the proposed methods, Cartesian grids are used, a variety of which are grids with boundary adaptation. The technology of constructing 3D vertically uniform boundary-adaptive grids on the basis of surface 2D grids, which is created using the procedure of minimizing the generalized Dirichlet functional, is presented. Previously, this approach has shown its effectiveness in constructing 2D non-degenerate regular grids containing the minimum number of cells (convex quadrangles) for test Z-shape regions, such as the “Maltese cross” and others, as well as in the 2Dhydrophysicsproblems numerical solution of coastal systems.


Introduction
The article discusses 3D diffusion-convection-sedimentation model of suspended matter in the aquatic environment, taking into account the coastal system features. Convective diffusion transfer operators in horizontal and vertical directions for the suspension transport problems have significantly different physical and spectral properties [1][2][3][4]. In addition, real coastal systems have complex coastline shape and bottom topography. This geometry of the computational domain motivates the boundary-adaptive grid construction [5 -6].
To construct effective algorithms for the suspended matter distribution operational forecast (reservoir pollution) in coastal systems that involve the multiprocessor computing systems use, it is advisable to use locally two dimensional splitting schemes (LTDS) [7]. Earlier it was shown for spatially 3D initial-boundary value problems for parabolic type equations that the LTS use makes it possible to obtain low-cost algorithms for the number of arithmetic operations for solving parabolic grid equations, which are economical in terms of the operations number and the time required for exchanges between processors [8].
To ensure regularity and exchanges simplicity, it is desirable to build 3D region boundary-adaptive grid as a Cartesian product of optimal 2D boundary-adaptive grid [9,10], built on the surface of the reservoir and uniform along the vertical direction. This is most easily achieved by replacing the Cartesian coordinate  varying everywhere in a closed interval   0;1 .
After changing the variables and constructing the grid, additive scheme for splitting the original problem along geometric directions is constructed. A chain of two problems is formulated, the first of which contains the 2Ddiffusion transfer operator and the convective transfer operator written in symmetric form, and the second problem contains the convective transfer operator in the vertical direction taking into account gravitational settling symmetrical formand the diffusion type operator. Then discretization is performed by the finite volume method.  In the model, carry out the transition from zcoordinate system to thecoordinate system, for which we use the Cartesian coordinate system in the horizontal plane and as the vertical variable -the dimensionless variable   , 0;1.


In the coordinate system, the water column is divided into the same number of layers at each point regardless of the depth, therefore, when using the «new» coordinate system, some problems associated with adding and subtracting layers are solved [11,12].
At θ transition tocoordinate system use the formula: .
The equation describing the behavior of the particles will look like this: Let us add the initial and boundary conditions to Eq. (3) assuming that the deposition of particles to the bottom is irreversible): -initial conditions at 0 t -boundary conditions at the lateral boundary S at any time where n is the outward normal to the boundary of the region Assuming that the bottom surface is not «strongly» curved.
In [13,14], the conditions for the correctness of the problem of transport of suspensions were investigated in the case of a multicomponent particle size distribution of particles, and therefore problem (3) -(8), as its particular case under the smoothness conditions of the solution function and the required smoothness of the region boundary.

Technology for constructing three-dimensional boundaryadaptive grids description
On the undisturbed surface of the reservoir , , 1, , 1, , by the given coordinates of the boundary nodes. One- built by volume V and has uniform steps within the area. The desired grid  find as the Cartesian product of a 2D grid 12  and onedimensional grid 3 Let's take a closer look at the construction of boundary-adaptive 2D grids.
In the plane   For convenience of perception, it is advisable to present a local consideration of the grids by considering the corresponding cells 12 12 , . Set the nodes of the cell 12  from coordinates The cells are convex quadrangles, where each of the corners is less than  , which ensures that the Jacobian is positive (9) where  is the iteration parameter, 1   . Note that a necessary condition for the minimum of the functional is a consequence of the conformity of the transformation under consideration.
The minimization h I on a set of convex grids was carried out by means of a quasi-Newtonian procedure in the computational domain   ,  . To do this, it is advisable to use the well-proven gradient methods tested and described in [11] in relation to the Azov Sea and the Taganrog Bay.
A change in the sign in the denominators of the integrands of the functionals (13), (14) of the Jacobian J , leads to discontinuities of the second kind, which turns out to be critical 6 E3S Web of Conferences 224, 02017 (2020) TPACEE-2020 It is important to note that when generating the grid using the above procedure, an initial approximation in the form of a convex grid is given. If this is not the case, then the initial grid, a convex grid, is determined by sequentially solving optimization problems.
Note that since cells with numbers containing 12 1, , 1, i N j N  are boundary, then the position of the nodes is not optimized for them. Therefore, further consideration of the problem is carried out on the nodes 12 2, 1, 2, 1 For a clear presentation of further reasoning, we represent (14) in the form Let introduce the notation   2 2 1 ,, 1 Taking into account (18), expression (17) can be compactly represented in the following form , It turns out that the constructed LDS is preferable for a moderate and large number of processors in comparison with a full approximation scheme and a splitting scheme in spatial variables using one-dimensional difference problems (local one-dimensional problems).

Conclusion
A model of 3D diffusion-sedimentation of suspended matter in an aqueous medium was considered, taking into account the proven system, for which economical algorithms for the operational forecast of impurity propagation were built on the basis of a local twodimensional splitting scheme. In this case, special attention is paid to the construction of a high-quality computational grid. The technology of constructing three-dimensional vertically uniform boundary-adaptive grids based on surface two-dimensional grids, when creating a generalized Dirichlet functional, is presented. The results of numerical test experiments on the construction of three-dimensional optimal boundary-adaptive grids are presented.