Accounting Method of Filling Cells for the Solution of Hydrodynamics Problems with a Complex Geometry of the Computational Domain

Abstract

This article is devoted to the development and application of the filling cells method for the solution of hydrodynamics problems with a complicated geometry of the computational domain, in particular, a liquid domain, to increase the smoothness and accuracy of the finite-difference solution. The spatial-two-dimensional flow problem of a viscous fluid between two coaxial semicylinders 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 are used to split 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. An analytical solution describing the Taylor-Couette flow is used as the reference to assess the accuracy of the numerical solution of the first problem. The simulation is performed on a a sequence of condensing computational grids with the following dimensions: 11 × 21, 21 × 41, 41 × 81, and 81 × 161 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 us to reduce the error to 6%. It is shown that splitting up the rectangular grid by factors of 2 to 8 in each of the spatial directions does not lead to the same increase in the accuracy of the numerical solutions obtained taking into account the filling of the cells.