A Method for Solving Grid Equations for Hydrodynamic Problems in Flat Areas

This paper discusses the numerical implementation of the mathematical model of the hydrodynamic process in the computational domain with an “extended geometry,” when its characteristic dimensions in the horizontal direction significantly exceed the vertical dimension. This is the typical property of a shallow water body or coastal system, which requires the development of specialized methods for solving the problems that arise in the process of discretizing grid equations. The explicit-implicit scheme has proved itself effective in solving the problem of transport in a shallow water body. The transition between time layers can be considered as an iterative process for solving the problem of diffusion-convection to settle. This idea forms the base for the formation of a preconditioner in the proposed method for solving grid equations obtained by approximating hydrodynamic problems in areas with extended geometry. A numerical experiment is carried out with the developed software module, which makes it possible to estimate the norm of the residual vector obtained by solving the grid equations of the pressure calculation problem based on the modified alternating triangular method (MATM) and the method for solving grid equations with a tridiagonal preconditioner, taking into account the hydrostatic approximation. According to the specifics of the developed method, it is effective in solving problems of aquatic ecology in the case of the computational domain, when its horizontal dimensions significantly exceed the vertical dimensions.