Parallel numerical method for solution of hydrodynamic equations in the shallow water approximation for shared memory computers

Modeling of natural phenomena such as tsunamis, floods, ocean and river currents, and dam breaks are pressing environmental problems. Due to the significant difference in the vertical and horizontal dimensions of the study area, non-stationarv two-dimensional hydrodynamic equations in the shallow water approximation have become very popular among researchers of the processes under consideration using mathematical modeling methods, which makes it possible to significantly simplify the mathematical formulation of the problems under consideration. Note, that the accuracy of numerical prediction using such equations depends on a number of conditions, among which the main ones are the spatial resolution of the study area and the quality of the selected numerical methods. However, increasing the spatial resolution significantly increases the time required to obtain a numerical solution, since in the numerical modeling of processes in the environment, as a rule, explicit or semi- implicit difference schemes are used with a limitation on the time step, depending on the size of the spatial grid steps. That is, as a result, due to the need to carry out calculations with a large number of grid nodes and a smaller time step, calculations on a computer with a sequential architecture can take quite a long time. The aim of this work is to formulate a mathematical problem of unsteady isothermal turbulent flows in river, based on the shallow water approximation, to construct an effective conservative numerical method, and to develop parallel computing algorithms for multi-core computing systems with shared RAM. The turbulent isothermal motion of an incompressible Newtonian fluid in a river flow is considered. The study area is an open river channel with islands and irregular bottom topography. Current characteristics can vary significantly over time, and therefore the current is considered unsteady with potential flooding of coastal areas. The movement of water in a river is determined by the forces of gravity and friction. In addition, the influence of the Coriolis force and turbulent diffusion on the flow is taken in to account. It is assumed that the horizontal dimensions of the region significantly exceed the vertical ones, the Reynolds-averaged flow characteristics change slightly in the vertical direction, and the vertical pressure distribution is hydrostatic. The thermophysical properties of water (viscosity, density) are considered constant. For numerical modeling of unsteady isothermal turbulent flows in river flows, a mathematical model is formulated based on the shallow water approximation for the Reynolds equations for an incompressible fluid. To take into account the transport, generation, diffusion and dissipation of turbulent vortices, this work uses the к - e turbulence model constructed by Rastogi and Rodi from the original k - e model of Launder and Spalding to close the Reynolds shallow water equations. To construct a discrete analogue of the developed mathematical model, the rectangular computational domain in which the flow with variable boundaries is studied is covered with a structured mesh with steps Аж, Ay, respectively. According to the concept of the finite volume method, each internal mesh node appears in a separate finite volume. In this case, the values of the flow depth, bottom topography (and turbulence model parameters) are determined at the nodes of the computational grid, and the velocity vector components are determined at the midpoints of the corresponding faces of the finite volumes. The differential equations of the model are integrated over each finite volume. The equations of motion are approximated in cells shifted by Аж/2, Ay/2, respectively. For approximate calculation of integrals, quadrature formulas of average rectangles are used, derivatives are approximately calculated using central-difference formulas. To calculate fluxes on the faces of finite volumes, monotonized upstream van Leer approximations [1] with a limiter [2] are used. When approximating equations in time, conditionally stable semi-implicit difference schemes are used, which are conservative not only for the continuity equation, but also for the equations of motion, which is very important for obtaining non-negative values of flow depth [3]. As a result, semi-implicit difference schemes of first order approximation in time and second order in spatial coordinates are obtained. The classical two-dimensional Thacker problem of fluid oscillations without friction in a reservoir whose bottom is a paraboloid was considered as a problem on which the accuracy of calculations and the quality of parallelization were tested. Satisfactory agreement between the calculated values of the flow depth and the analytical solution was obtained.