The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method.

The solution to equations of two viscous homogeneous incompressible fluid media with the pressure phase equilibrium in the case of a constant phase is obtained. The influence of the physical phase densities, saturation, volume and viscosity of substances constituting a two-phase continuum in the flow velocity and pressure is shown. Also, the solution admitting a limiting transition to the known solution of the problem of a flow of a viscous incompressible single-phase medium is constructed.

The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that: 1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region; 2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC; 3) the proposed computational algorithm is stable; 4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable); 5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time. Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.

Chemical compositional simulation of enhanced oil recovery and surfactant enhanced aquifer remediation processes is a complex task that involves solving dozens of equations for all grid blocks representing a reservoir. In the present work, we perform a numerical validation of the newly developed mathematical formulation which satisfies the conservation laws of mass and energy and allows applying a sequential solution approach to solve the governing equations separately and implicitly. Through its application to the numerical experiment using a wettability alteration model and comparisons with existing chemical compositional model’s numerical results, the new model has proven to be practical, reliable and stable.

This article presents the continuation problem for one-dimensional equations of acoustics. Currently, one of the most difficult areas of research in applied mathematics is the reverse calculation of wave propagation, which has an important area of application in geophysics and medicine. The propagation of an acoustic wave in an inhomogeneous medium and the measurement of the return of some part of the sound by special devices leads to the calculation of the initial-boundary for the partial derivative equation. Most of these problems are ill-posed. One such ill-posed problem is the continuation problem. The continuation problem is based on finding the value of the desired function in the rest of the boundary using additional data in a certain part of the boundary. In this regard, we present the continuation problems to the inverse boundary value problem. There are many methods for solving general boundary value inverse problems, for example, the gradient method. In this paper, we construct a finite-difference scheme for this inverse problem and find an unknown function on the characteristic from this difference equation by inverting the difference scheme. The effectiveness of this method lies in a simple and accurate fast calculation algorithm. To test the correctness of any method or algorithm, we first solve a direct problem by specifying a exact function. Then we find additional information and solve the inverse problem. Let us compare the inverse problem with solving a problem with an exact function. However, in practice it often happens that additional information is provided with certain errors. Therefore, the correctness of the algorithm was verified by adding different error levels to the additional information. Quantitative data and graphs of these quantitative results are compared.