Parallel CUDA implementation of a numerical algorithm for solving the Navier-Stokes equations using the pressure uniqueness condition

Abstract

In this paper, we study numerical methods for solving the Navier-Stokes equations in doubly connected domains. Two methods for solving the problem are considered. The first method is based on constructing a difference problem in variables of the stream function and the vortex of velocity using the uniqueness condition for pressure. The numerical solution of the elliptic equation for stream functions is found as the sum of the solutions of two simple problems of an elliptic type. One problem is with homogeneous boundary conditions, and the other is with a homogeneous equation. An alternative approach to solving the problem is the fictitious domain method with the continuation of the least coefficient. This method does not require satisfying the pressure uniqueness condition, and is simple to implement. An important direction in the development of numerical simulation methods is the study of approximate methods for solving problems of mathematical physics in complex multidimensional areas. To solve many applied problems in irregular areas, the fictitious domain method is widely used, which is characterized by a high degree of automation of programming. The main idea of the fictitious domain method is that the problem is solved not in the original complex domain, but in some other, simpler domain. This allows to create software immediately for a fairly wide class of problems with arbitrary computational domains. The possibilities of applying the fictitious domain method to the problems of hydrodynamics in the variables “stream function, vortex of velocity” are considered in many works. In this paper, we study a numerical method for solving the Navier-Stokes equations in doubly connected domains. A computational finite difference algorithm for solving an auxiliary problem of the fictitious domain method has been developed. The results of numerical modeling of the two-dimensional Navier-Stokes equations by the fictitious domain method with continuation by the lowest coefficient are presented. For this problem, a parallel algorithm was developed using the CUDA architecture, which was tested on various grid dimensions.