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Том 64, № 6 (2024)
General numerical methods
RATIONAL ARITHMETIC WITH A ROUND-OFF
Аннотация
Computer calculations in floating-point arithmetic are always approximate. In contrast, calculations in rational arithmetic (for example, in computer algebra) are always absolutely precise and reproducible both on other computers and (theoretically) manually. Therefore, such calculations can be demonstrative in the sense that the proof obtained with their help is no different from the traditional one. However, such calculations are usually impossible in a sufficiently complex problem due to limited memory and time resources. We propose a mechanism for rounding off rational numbers in calculations in rational arithmetic, which solves this problem (of resources), i.e. the calculations can still be demonstrative, but no longer require unlimited resources. A number of examples of the implementation of standard numerical algorithms in this arithmetic are given. The results have applications to analytical number theory.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):895-913
895-913
ON THE ASYMPTOTICS OF EIGENVALUES OF SEMIDIAGONAL TOEPLITZ MATRICES
Аннотация
Asymptotic formulas are constructed that allow a uniform estimate of the remainder term for Toeplitz matrices of size 𝑛 for 𝑛 → ∞ in the case when their symbol 𝑎(𝑡) has the form 𝑎(𝑡) = (𝑡 − 2𝑎0 + 𝑡-1)3. This result is a generalization of the result of Stukopin et al. (2021), in which similar asymptotic formulas were obtained for a diagonal Toeplitz matrix with a symbol of a similar form when 𝑎0 = 1. The obtained formulas have high computational efficiency and generalize the results of the classical works of Parterre and Widom on the asymptotics of extreme eigenvalues.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):914-921
914-921
FORMULAS FOR NUMERICAL DIFFERENTIATION ON A UNIFORM GRID IN THE PRESENCE OF A BOUNDARY LAYER
Аннотация
The problem of numerical differentiation of functions with large gradients is considered. It is assumed that for the original function of one variable the decomposition is valid as the sum of a regular component with bounded derivatives up to a certain order and a boundary layer component having large gradients and known with an accuracy of up to a factor. Such a decomposition, in particular, is valid for solution of a singularly perturbed boundary value problem. The topic of the study is relevant, since the application of classical polynomial formulas of numerical differentiation to functions with large gradients can lead to significant errors. The error of the formulas of numerical differentiation, according to the construction of exact ones on the boundary layer component of the original function, is estimated. The results of numerical experiments are presented, consistent with the obtained error estimates.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):922-931
922-931
Optimal control
932-939
FAULT-TOLERANT FAMILIES OF PRODUCTION PLANS: MATHEMATICAL MODEL, COMPUTATIONAL COMPLEXITY AND BRANCH AND BOUND ALGORITHMS
Аннотация
The design of fault-tolerant production and supply systems is one of the priority areas of development of modern operations research. The traditional approach to modeling such systems is based on the use of probabilistic models describing the choice of a possible scenario of actions in the event of failures in the production or transport network. Along with a number of advantages, this approach has a wellknown drawback. The occurrence of failures of an unknown nature that can jeopardize the operability of the entire modeled system significantly complicates its application. In this paper, we introduce the minimax problem of constructing fault-tolerant production plans (Reliable Production Process Design Problem, RPPDP), the purpose of which is to ensure the smooth functioning of a distributed production system with minimal guaranteed costs. It is shown that the RPPDP problem is NP-hard in the strong sense and remains intractable under fairly specific conditions. To find exact and approximate solutions with accuracy estimates for this problem, branch and bound methods have been developed based on the proposed compact model of mixed integer linear programming (MILP) and the author’s heuristics of adaptive large neighborhood search (ALNS) within the framework of extensions of the well-known Gurobi MIP-solver. High performance and complementarity of the proposed algorithms have been confirmed by the results of numerical experiments conducted on an open library of test examples developed by the authors, containing adapted problem statements from the PCGTSPLIB library.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):940-958
940-958
ON THE CONTROLLABILITY OF DISTRIBUTED PARAMETER SYSTEMS
Аннотация
The problem of controllability for optimal control problems, optimization of systems with distributed parameters in partial derivatives is considered. The concept of controllability as correctness according to A. N. Tikhonov for solving optimization problems is introduced. A theorem with controllability conditions for direct solution (direct minimization of the objective functional) of optimization problems by extremal algorithms is given. A test example of numerical solution of the optimization problem for a nonlinear hyperbolic system describing non-stationary water flow in an open channel is considered. Controllability analysis is demonstrated, which ensures correctness of the solution of the problem and high accuracy of optimization of the distributed friction coefficient in the flow equations.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):959-972
959-972
Ordinary differential equations
ON THE APPROXIMATION OF THE FIRST EIGENVALUE OF SOME BOUNDARY VALUE PROBLEMS
Аннотация
The paper studies the representation of eigenfunctions as scalar series for a two-point boundary value problem of the type (𝑛 − 1, 1) under the assumption that there exists a functional concentrated at one point such that the first 𝑛−1 of the original boundary conditions and ℓ̃𝑥 = 1 become the Cauchy conditions at this point. The eigenfunction of the boundary value problem under consideration, corresponding to the eigenvalue λ∗, is represented as a series in powers of λ∗. The equation Φ(λ) = 0, where Φ(λ) is the sum of the series in powers of λ, is considered for finding the eigenvalues of the original problem. Examples of calculating the first eigenvalue of some boundary value problems are given. Various estimates are obtained for the coefficients of such power series. A certain function of two variables 𝑡 and λ is defined, a partial differential equation is obtained for it, and conditions are obtained that it satisfies. The zeros of the “section” of this function coincide with the eigenvalues of the original boundary value problem, which can be used for their approximate calculation.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):973-991
973-991
ANALYTICAL-NUMERICAL METHOD FOR SOLVING THE SPECTRAL PROBLEM IN A MODEL OF GEOSTROPHIC OCEAN CURRENTS
Аннотация
A new efficient analytical-numerical method is developed for solving a problem for the potential vorticity equation in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum. The method is used to analyze small perturbations of ocean currents of finite transverse scale with a general parabolic vertical profile of velocity. For the arising spectral nonself-adjoint problem, asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small wave numbers and the existence of a countable set of complex eigenvalues with an unboundedly decreasing imaginary part is shown. On the integration interval , a system of three neighborhoods is introduced and a solution in each of them is constructed in the form of power series expansions, which are matched smoothly, so that the eigenfunctions and eigenvalues are efficiently calculated with high accuracy. For a varying wave number, the trajectories of complex eigenvalues are computed for various parameters of the problem and the existence of double eigenvalues is shown. The complex picture of instability developing in the simulated flow depending on physical parameters of the problem is briefly described.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):992-1007
992-1007
EXISTENCE OF SOLUTIONS TO THE NON-SELF-ADJOINT STURM-LIOUVILLE PROBLEM WITH DISCONTINUOUS NONLINEARITY
Аннотация
The problem of existence of solutions of the Sturm-Liouville problem with a non-self-adjoint differential operator and non-linearity discontinuous in the phase variable is considered. Theorems on the existence of non-trivial (positive and negative) solutions for positive values of the spectral parameter are established for the problem under study. Examples illustrating the obtained theorems are given.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):1008-1015
1008-1015
Partial Differential Equations
1016-1027
ON INITIAL-BOUNDARY VALUE PROBLEMS FOR PARABOLIC SYSTEMS IN A SEMI-BOUNDED PLANE DOMAIN WITH GENERAL BOUNDARY CONDITIONS
Аннотация
The paper considers initial boundary value problems for homogeneous parabolic systems with Dini-continuous coefficients under zero initial conditions in a semi-bounded plane domain with a non- smooth lateral boundary that admits the presence of ”beaks”on which boundary conditions of a general type with variable coefficients are specified. Using the method of boundary integral equations, a theorem is proved on the unique classical solvability of such problems in the space of functions that are continuous and bounded together with their first-order spatial derivative in the closure of the domain. A representation of the solutions obtained is given in the form of vector single layer potentials.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):1028-1041
1028-1041
Mathematical physics
TURBULENT KINETIC ENERGY IN AN APPROXIMATE SOLVER OF THE RIEMANN GAS DYNAMICS PROBLEM
Аннотация
The paper describes the consideration of turbulent kinetic energy in solving the gas-dynamic problem of discontinuity decay (Riemann problem) using the HLLC approximate solver. The system of Euler equations is considered with the addition of the hyperbolic equation of turbulent kinetic energy and consideration of turbulent pressure in the momentum and energy balance equations. The Jacobian coefficient of the system of equations and its eigenvalues are found. Based on this, changes are made to the calculation scheme in the HLLC solver. Using the Sod problem as an example, the correctness of taking into account turbulent kinetic energy in solving the Riemann problem is verified, and the instability of the scheme at high turbulent pressure is shown in the case of not taking turbulence into account in calculating the characteristic velocities.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):1042-1054
1042-1054
MODELING OF ICE-WATER PHASE TRANSITION IN A PIPE WITH SMALL ICE BUILDUPS ON THE WALL
Аннотация
The mathematical modeling of the ice-water phase transition during fluid flow inside a pipe with a small ice buildup on the wall at high Reynolds numbers is considered. As a mathematical model describing the dynamics of the phase transition, a double-deck boundary layer model and a phase field system are used. The results of numerical simulation are presented.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):1055-1063
1055-1063
PROBLEMS OF DETERMINING QUASI-STATIONARY ELECTROMAGNETIC FIELDS IN WEAKLY INHOMOGENEOUS MEDIA
Аннотация
Statements of initial-boundary value problems for the system of Maxwell equations in various quasi-stationary approximations in homogeneous and inhomogeneous conducting media are considered. In the case of weakly inhomogeneous media, asymptotic expansions of solutions of the initial-boundary value problems under consideration in a parameter characterizing the degree of inhomogeneity of the medium are formulated and substantiated. It is shown that the construction of an asymptotic expansion for a quasi- stationary electromagnetic approximation leads to a sequential solution of independent problems for a quasi- stationary electric and quasi-stationary magnetic approximation in a homogeneous medium. Conditions on the initial data are given for which the asymptotic series are convergent.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):1064-1081
1064-1081
NUMERICAL SIMULATION OF CONVECTIVE FLOWS IN A THIN LIQUID LAYER UNDER CONDITIONS OF LARGE REYNOLDS NUMBERS
Аннотация
A mathematical model is proposed that describes the flow of a thin layer of liquid on an inclined, non-uniformly heated substrate. The Navier-Stokes system for a viscous incompressible liquid and relations representing generalized kinematic, dynamic and energy conditions at the interface for the case of evaporation are used as governing equations. The statement is given in a two-dimensional case for large Reynolds numbers. The problem is solved within the framework of the long-wave approximation. A parametric analysis of the problem is carried out, an evolutionary equation is obtained for finding the thickness of the liquid layer. An algorithm for a numerical solution is proposed for the problem of periodic flow of liquid down an inclined substrate. The influence of gravitational effects and the nature of heating of a solid substrate on the flow of a liquid layer is studied.
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki. 2024;64(6):1082-1094
1082-1094