Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki

This work investigates a problem for a fractional diffusion equation with nonclassical boundary conditions. A family of weighted difference schemes is studied for the considered problem. An algorithm for finding a numerical solution is provided. Using the maximum principle for the difference problem, an a priori estimate is derived, which implies the stability of the difference schemes and the convergence of the numerical solution to the exact solution in the C-norm.

The interpolation and mosaic-skeleton methods for solving the problem of potential flow of a two-dimensional plate are compared. They compress the dense matrix of the linear system arising from the solution by the collocation method on an irregular grid. The first method is based on fast Fourier transform and linear interpolation with an auxiliary uniform grid. The second one is based on block-majorange approximation of the matrix. Both approaches demonstrate time and memory efficiency, but emphasize different structures in the matrix, which affects the solution of the linear system. For the utilized implementations of the mosaic-matching methods The skeleton method solves the system faster than the interpolation method, but consumes more memory, and its running time grows much more noticeably as the size of the system increases.

The paper deals with a noniterative overlapping domain decomposition methods for solving multidimensional parabolic problems. The structure of the method is similar to that of a component-by-component splitting scheme using smooth unit partitioning. The method itself has been known for a long time, but a number of important technical details related to the design of partitioning a domain into subdomains that provides smooth unit partitioning have not been published previously. In this paper, these details are given as a set of provable assertions. A process for obtaining an error estimate is described, the details of which have also not been previously published.

The concepts of chaotic systems and systems of differential equations with impulse action are formulated. Chen’s chaotic system is considered. It is a system of three ordinary differential equations admitting a zero solution which is Lyapunov unstable. The problem of stabilization of this solution is posed. This problem is solved by two methods: by impulsive actions at fixed points in time and by impulsive actions that occur on some set of phase space.

We consider algorithms for calculating the spread of epidemics and analyzing the consequences of introducing or removing restrictive measures based on the SIR model and the Hamilton–Jacobi–Bellman equation. After studying the identifiability and sensitivity of the SIR models, the correctness in the neighborhood of the exact solution and the convergence of the numerical algorithms for solving forward and inverse problems, the optimal control problem is formulated. Numerical simulation results show that feedback control can help determine vaccination policies. The use of PINN neural networks reduced the computation time by a factor of 5, which seems important for promptly changing restrictive measures.

A quasilinear hyperbolic equation is considered whose principal part is a purely wave operator and the lower order part consists of two nonlinear terms with coefficients and compactly supported in a ball. We study the direct problem of a plane wave scattered by a heterogeneity localized in and the inverse problem of recovering the coefficients and from solutions of direct problems with a varying incident wave direction. An asymptotic expansion of the solution to the direct problem near the front of the traveling plane wave is presented, based on which the inverse problem is reduced to two linear problems to be solved sequentially. Namely, the problem of determining the coefficient is reduced to a classical X-ray tomography problem, while the problem of determining the coefficient is reduced to a more complicated problem of integral geometry. The last problem, which is new, is to find a function from its integrals with a given weight along straight lines. This problem is investigated, and a uniqueness and stability theorem for its solution is proved.

This paper presents the current state of research in the field of variational assimilation of observational data in problems of geophysical hydrodynamics, developed by G.I. Marchuk and his scientific school for many years. For the ocean dynamics model developed at the IWM RAS, we present a technology of four-dimensional variational assimilation of observational data (4D-Var) based on a combination of splitting and conjugate equation methods. The technology includes minimization of the cost functional describing the difference between the model solution and observational data with covariance matrices of observation errors and initial approximation. Application of the multicomponent splitting method makes it possible to solve the system of direct and coupled equations stepwise. Effective algorithms for solving variational problems of data assimilation on the basis of modern iterative processes with a special choice of iterative parameters are proposed. Developing G.I. Marchuk's idea of searching for energy-active zones in the ocean that determine its heat exchange with the atmosphere, new algorithms are developed to investigate the sensitivity of the model solution to errors in observational data. The methodology is illustrated for a model of the Black Sea hydrothermodynamics with variational data assimilation to recover heat fluxes at the sea surface. The conclusion discusses the prospects for the development of this direction.

In this paper, we investigate the dissipative-dispersive properties of projection- and of the characteristic method of the third order of approximation for numerical solution of the advection equations. This scheme is called Cubic Polynomial Projection, and it is constructed by a grid-based scheme by the characteristic method using Hermite interpolation. The properties of this scheme are compared with similar properties of the Cubic Interpolation Polynomial scheme, widely used in computational practice and also based on Hermite interpolation. Both schemes belong to the class of characteristic schemes, which is important for particle transport problems and explicit consideration of the exponential dependence of the solution on the optical thickness. Instead of the traditional interpolation closure characteristic of the Cubic Interpolation Polynomial scheme, the Cubic Polynomial Projection scheme uses orthoprojector closure. This allows to transfer this scheme to unstructured tetrahedral meshes and solves the problem of coplanarity of the characteristic of one of the faces of the cell, but doubles the required memory resources in the simplest one-dimensional case. The paper shows that projective closure significantly improves the already quite good dissipative-dispersive properties of the Cubic Interpolation Polynomial scheme, significantly approaching them to the dissipative-dispersive properties of the exact solution of the advection equation. These conclusions are confirmed by numerical calculations.

The article develops a mathematical model for burning coke sedimentations from a layer of an aluminosilicate cracking catalyst with a spherical grain, taking into account heterogeneous detailed chemical reactions. The model is a system of equations of mathematical physics with initial and boundary conditions. An explicit-implicit computational algorithm has been developed for the constructed mathematical model. A comparison of the calculation results with experimental data on the material balance and theoretical estimates for temperature revealed the adequacy of the mathematical model and computational algorithm.