Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
ISSN (print): 0044-4669
Founders: Russian Academy of Sciences, Federal Research Center IU named after. A. A. Dorodnitsyna RAS
Editor-in-Chief: Evgeniy Evgenievich Tyrtyshnikov, Academician of the Russian Academy of Sciences, Doctor of Physics and Mathematics sciences, professor
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卷 65, 编号 6 (2025)
K stoletiyu so dnya rozhdeniya G.I. Marchuka
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):841
841
General numerical methods
ON THE DIFFERENCE SOLUTION OF ONE NONLOCAL BOUNDARY VALUE PROBLEM FOR THE FRACTIONAL ORDER DIFFUSION EQUATION
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):842-849
842-849
DECOMPOSITION-COMPOSITION SCHEMES FOR SYSTEMS OF SECOND-ORDER EVOLUTIONARY EQUATIONS
摘要
We consider numerical methods for approximate solution of the Cauchy problem for coupled systems of second-order evolution equations. Simplification of the problem at a new layer in time is achieved by separating simpler subproblems for the individual components of the solution. The computational technique of decomposition-composition consists of two steps. First, the decomposition of the operator matrix of the problem is performed, and then an approximate solution is constructed based on a linear composition of the solutions of the auxiliary problems. In this paper, we investigate decomposition variants based on the extraction of the diagonal part, lower and upper triangular submatrices of the operator matrix, as well as when splitting the operator matrix into rows and columns. Different variants of splitting schemes are used at the composition stage. In two-component decomposition, explicit-implicit schemes and factorized schemes are distinguished. Regularized additive schemes are used in multi-component splitting. The study of stability of three-layer decomposition-composition schemes is carried out on the basis of stability theory of operator-difference schemes in finite-dimensional Hilbert spaces.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):850-860
850-860
COMPARISON OF INTERPOLATION AND MOSAIC-SKELETON METHODS FOR SOLVING INTEGRABLE EQUATIONS WITH CONVOLUTIONAL KERNEL
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):861-874
861-874
MULTIGRID METHODS OF MACRO-GRID DOMAIN DECOMPOSITION
摘要
We consider integrated multigrid domain decomposition methods (DDM-MG) for solving large systems of linear algebraic equations (SLAEs) with sparse symmetric or asymmetric matrices and multivariate boundary value problems obtained by grid approximations. The proposed algorithms are based on the construction of single-layer or two-layer macrogrids and special ordering of nodes according to their belonging to different topological primitives of the macrogrid: macro nodes, macro edges, macro faces and subareas. At coordinated numbering of vector components, the SLAU matrix in the three-dimensional case takes a block-tri-diagonal form of the fourth order. For its solution we use some method of approximate filtering in Krylov subspaces. At the same time, the solution of auxiliary systems in subspaces is carried out by multigrid methods of block incomplete factorization, on the basis of similar topology-oriented ordering of nodes, but not at the macro-, but at the micro-level, resulting in the formation of a single preconditioner of recursive-nested type. The justification of the proposed methods is carried out for Stiltjes-type matrices.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):875-887
875-887
ON SOME APPROACHES TO DOMAIN DECOMPOSITION METHODS FOR SOLVING PARABOLIC PROBLEMS
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):888-906
888-906
Optimal control
ON IMPULSE STABILIZATION OF CHAOTIC CHEN SYSTEM
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):907-917
907-917
Optimal Disturbances of Stationary and Periodic Solutions to Delay Systems in Mathematical Immunology
摘要
This work is devoted to optimal disturbances of stationary and periodic solutions to systems of delay differential equations, their computation, and use in mathematical immunology. Original methods for computing the stationary and periodic solutions themselves and tracing them along the system parameters, as well as methods for computing optimal disturbances for these solutions are briefly described. The performance of the described methods is demonstrated using the example of the well-known Marchuk–Petrov model of the antiviral immune response with parameter values corresponding to the infection caused by hepatitis B viruses.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):918-945
918-945
Ordinary differential equations
DIFFERENTIAL EPIDEMIC MODELS AND SCENARIOS FOR RESTRICTIVE MEASURES
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):946-960
946-960
Partial Differential Equations
INVERSE PROBLEM FOR THE QUASILINEAR WAVE EQUATION
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):961-971
961-971
ON THE UNAMBIGUOUS SOLVABILITY OF INITIAL BOUNDARY VALUE PROBLEMS FOR PARABOLIC SYSTEMS IN A SEMI-BOUNDED FLAT REGION WITH A NONSMOOTH LATERAL BOUNDARY
摘要
We consider the initial boundary value problem for an inhomogeneous parabolic system with Dini-continuous coefficients with a nonzero initial condition in a semi-constrained flat region with a nonsmooth lateral boundary admitting beaks, on which boundary conditions of general form are set. We prove a theorem on the single-valued solvability of such a problem in the space of functions continuous and bounded together with their spatial derivative of the first order in the closure of the region. An integral representation is given and the smoothness character of the obtained solution is investigated.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):972-984
972-984
Mathematical physics
METHODS OF VARIATIONAL ASSIMILATION OF OBSERVATION DATA IN PROBLEMS OF GEOPHYSICAL HYDRODYNAMICS
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):985-998
985-998
TRANSPARENT BOUNDARY CONDITIONS FOR THE WAVE EQUATION WITH VARIABLE SPEED OF SOUND
摘要
A method of constructing a transparent boundary condition operator for a wave equation with variable sound velocity in a channel of rectangular cross section is proposed. A numerical example showing the performance of the proposed method is given. Properties of images of the convolution kernel functions of transparent boundary conditions are analyzed, a method of constructing their rational approximation is proposed, and its numerical convergence is shown.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):999-1016
999-1016
DISSIPATIVE-DISPERSIVE PROPERTIES OF ONE PROJECTIVE METHOD FOR NUMERICAL SOLUTION OF THE ADVECTION EQUATION
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):1017-1028
1017-1028
ADAPTATION OF THE FINITE ELEMENT METHOD FOR THE STIELTJES STRING DEFORMATION PROBLEM WITH A NONLINEAR CONDITION
摘要
We study a problem modeling small deformations of a string with features localized in an arbitrary number of points (but not more than a countable number) in the form of elastic supports and concentrated forces. It is assumed that the left end of the string is rigidly fixed and the right end is inside a vertical displacement limiter. Depending on the applied external force, the right end will either remain free or reach the boundary of the limiter. This generates a nonlinear condition at the corresponding point, since the behavior of the solution is not known in advance. The problem under study is described in the form of a variational inequality; the existence and uniqueness theorems of the solution are proved; an algorithm for finding an approximate solution is developed by adapting the finite element method; and an estimate of the deviation of the exact solution from the approximate solution is obtained.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):1029-1044
1029-1044
MATHEMATICAL MODELING OF UNSTEADY COMBUSTION OF COKE SEDIMENTATIONS IN A CATALYST LAYER WITH A SPHERICAL GRAIN
摘要
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.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2025;65(6):1045-1056
1045-1056