Accounting method of filling cells for the hydrodynamics problems solution with complex geometry of the computational domain

The article is devoted to the development and application of the filling cells method for the hydrodynamics problems solution with complicated geometry of the computational domain, in particular, liquid, to increase the smoothness and accuracy of a finite-difference solution. The spatial-two-dimensional flow problem of a viscous fluid between two coaxial semi-cylinders and the spatial-three-dimensional problem of wave propagation in the coastal zone demonstrate the possibilities of the proposed method. The rectangular grids are used to solve these problems, taking into account the filling of cells. The approximation of problems have been used splitting schemes in time for physical processes and the approximation in spatial variables is made using the balance method, taking into account the filling of cells and without it. Analytical solution describing the Taylor–Couette flow is used as a reference to assess the accuracy of the numerical solution of the first problem. The simulation was performed on a sequence of condensing computational grids with the following dimensions: $11\times21$, $21\times41$, $41\times81$ and $81\times161$ nodes in the case of using the method and without using it. In the case of the direct use of rectangular grids (stepwise approximation of boundaries), the relative error of the calculations reaches $70\%$; under the same conditions, the use of the proposed method allows to reduce the error to $6\%$. It is shown that splitting up rectangular grid by $2$–$8$ times in each of the spatial directions does not lead to the same increase of the numerical solutions accuracy, obtained taking into account the filling of the cells.