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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.

In this paper, we study an initial boundary value problem for a two-dimensional differential equation with a fractional time derivative in the sense of Caputo. This equation is of great applied importance in modeling flow processes and anomalous dispersion. The uniqueness and continuous dependence of the solution on the input data in differential form is proved. A computationally effective implicit scheme with weights is proposed. A priori estimates are obtained for the solution of the problem under the assumption that a solution exists in the class of sufficiently smooth functions. These estimates imply the uniqueness of the solution and the stability of the scheme with respect to the initial data and the right-hand side of the equation. The convergence of the approximate solution to the solution of the differential problem with the second order both in time and space variables is proved. The results of computational experiments confirming the reliability of theoretical analysis are presented.

The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

The paper studies a model of the gaslift process where the motion in a gas-lift well is described by partial differential equations. The system describing the studied process consists of equations of motion, continuity, equations of thermodynamic state, and concentration equation. A finite-difference scheme is constructed for the numerical solution of the problem. Because of the complexity of the motion equation, two simplified difference analogues for this equation are considered. The stability of these equations are investigated using the method of a priori estimates. The estimates obtained will be used for proving the stability of the whole difference scheme in future works.

The paper focuses on constructing and investigation of cost-effective difference schemes for the numerical solution of the two-dimensional three-phase non-isothermal flow problem without capillary and gravitational forces. In this paper, the finite difference method is used to solve the problem numerically. The method of energy inequalities is applied to examine the stability of the finite difference scheme with respect to the initial data and the right-hand sides of the equations. Three cost-effective difference schemes are constructed on the base of the studied scheme. The efficiency of the proposed algorithms is analyzed on the basis of comparing the average time spent on the numerical implementation of one time layer.

The paper focuses on the numerical implementation of a model optimal control problem governed by equations of three-phase non-isothermal flow in porous media. The objective is to achieve preassigned temperature distribution along the reservoir at a given time of development by controlling mass flow rate of heat transfer agent on the injection well. The problem of optimal control is formulated, the adjoint problem is presented, and an algorithm for the numerical solution is proposed. Results of computational experiments are presented for a test problem.