Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Том 61, № 7 (2025)
NUMERICAL METHODS
On the construction of difference schemes for calculating viscous gas flows in orthogonal curved coordinates
Аннотация
A family of conservative difference schemes for calculating viscous gas flows in arbitrary orthogonal curved coordinates is constructed on the basis of a cartesian Godunov-type difference scheme of a general type. At the same time, the algorithms for calculating the numerical flows of the basic scheme remain unchanged. It is shown that the schemes of the constructed family have a second order of approximation in space, provided that the numerical flows of the basic scheme are calculated with a second or higher order of accuracy.
Differential equations. 2025;61(7):867–881
867–881
OPERATOR-DIFFERENCE SCHEMES FOR SYSTEMS OF FIRST-ORDER INTEGRO-DIFFERENTIAL EQUATIONS
Аннотация
The Cauchy problem is considered for a system of two first-order integro-differential equations with memory in finite-dimensional Hilbert spaces, where the integral term contains a difference kernel. Such mathematical model is typical for nonstationary electromagnetic processes, taking into account the dispersion effects of the electric field. To obtain an approximate solution to the considered nonlocal problem, a transformation to a local Cauchy problem for a system of first-order equations is applied, based on approximating the difference kernel by a sum of exponentials. Two-level operator-difference schemes in Hilbert spaces are constructed and analyzed for stability.
Differential equations. 2025;61(7):882–891
882–891
MATHEMATICAL MODELING OF THE MOTION OF METAL CONDUCTORS IN AN ELECTROMAGNETIC FIELD TAKEN INTO ACCOUNT OF THE PRESENCE OF DIFFERENT PHASES OF THE ACCELERATED MATTER
Аннотация
The problem of mathematical modeling of the acceleration of metal conductors in an electromagnetic field in a two-dimensional approximation has been solved. Mathematical models are presented to describe the motion of bodies using Lagrangian and Eulerian coordinates using the constitutive relations of a thermoelastoplastic body (for the case of large deformations) and a viscous compressible fluid (gas). A mathematical model is presented that allows us to describe the movement of a body taking into account the presence of different phases of matter in it at one point in time. The model explicitly identifies the transition phase from solid to liquid; for this phase, both constitutive relations are taken into account, taken with appropriate weights. Numerical algorithms based on the finite element method have been constructed. The presented model is used to solve the problem of accelerating an aluminum cylindrical shell to a velocity of about 8 km/s. The calculation results are demonstrated, and individual characteristics are compared with known calculated and experimental results.
Differential equations. 2025;61(7):892–909
892–909
IDENTIFICATION OF THE ORDER OF FRACTIONAL DERIVATIVE IN WINDKESSEL MODEL
Аннотация
We investigate windkessel blood flow model with fractional derivative. A cost-effective numerical ap- proximation of the model equation is considered, which allows calculations with high precision. The approximation is tested on the proposed special case with the existing analytical solution. We use pro- posed numerical approximation to test various methods to identify the fractional order from real blood pressure profiles. The obtained methods allow to determine the order of the fractional derivative with an accuracy not worse than 15 %.
Differential equations. 2025;61(7):910–918
910–918
A FULLY CONSERVATIVE FINITE DIFFERENCE SCHEME FOR THREE-DIMENSIONAL NAVIER–STOKES EQUATIONS IN CYLINDRICAL COORDINATES
Аннотация
The fully conservative finite volume discretization of the incompressible Navier–Stokes equations in cylindrical coordinates is constructed on a staggered grid. The proposed discretization ensures momentum conservation in a computational domain, and mass conservation within the control volumes for pressure, and velocity components. The energy conservation equation directly follows from the discrete momentum equation. Both conservative and non-conservative forms of convective terms are approximated. The proposed discrete counterpart of the vector Laplace operator is self-adjoint and negative definite.
Differential equations. 2025;61(7):919–940
919–940
NUMERICAL SOLUTION OF THE INVERSE COEFFICIENT PROBLEM FOR A MATHEMATICAL MODEL OF DESORPTION DYNAMICS
Аннотация
An inverse coefficient problem for a mathematical model of desorption dynamics is considered. The inverse problem is reduced to a nonlinear operator equation for an unknown coefficient. The operator equation is used to construct an iterative numerical method for solving the inverse problem. To prove the convergence of the iterative method, the contraction mappings principle is used. Examples of the application of the iterative method for the numerical solution of the inverse problem are given.
Differential equations. 2025;61(7):941–951
941–951
NUMERICAL STUDIES OF TWO-PHASE HYPERELASTIC MODEL
Аннотация
The paper is devoted to numerical studies of two-phase hyperbolic model describing the dynamics of hyperelastic media. The considered model is a generalization of the well known model of multivelocity fully nonequilibrium Baer–Nunziato model, widely used for description of shock-wave and detonation processes in multiphase media. The equations of the model are given both in the general and in the spatially one-dimensional case, and its properties are described. For numerical study, the pathconservative HLL method is applied. The numerical study is carried using a number of Riemann problems.
Differential equations. 2025;61(7):952–970
952–970
GRID-CHARACTERISTIC TWO-DIMENSIONAL SCHEMES FOR DYNAMIC PROBLEMS OF LINEARLY ELASTIC LAYERED MEDIA
Аннотация
The paper considers the stress-strain state of a layered geological medium under the influence of an external dynamic load. Each layer is described by an isotropic linear elastic model with specified mechanical parameters. For the numerical simulation of the wave propagation process in a two-dimensional problem formulation, a grid-characteristic scheme of a high approximation order was constructed. The issues of approximating boundary and contact conditions, the problem of the accuracy reduction for spatial splitting schemes are addressed. The numerical solution results for test problems are presented.
Differential equations. 2025;61(7):971–985
971–985
INTERNAL ESTIMATION OF THE INFORMATION SET OF THE PROBLEM OF PARAMETRIC IDENTIFICATION OF DYNAMIC SYSTEMS BASED ON INTERVAL DATA
Аннотация
The paper considers the method of internal interval estimation of the information set of the problem of parametric identification of dynamic systems, when the experimental data are specified in the form of intervals. The state of the dynamic systems under consideration at each moment of time is a parametric set. The objective function is constructed in the space of interval estimates of parameters, characterizing the degree of inclusion of parametric sets of states in the specified experimental interval estimates of phase variables. An expression for the gradient of the objective function is obtained. The proposed approach consists of two stages. At the first stage, the objective function is minimized by first-order optimization methods, and at the second stage, the obtained estimate of the information set is successively expanded with control of the value of the objective function. To solve a variety of direct problems in the process of constructing the desired estimate, the adaptive interpolation algorithm previously developed by the authors is used. The efficiency and performance of the approach under consideration is demonstrated on a representative series of problems.
Differential equations. 2025;61(7):986–999
986–999
FINITE DIFFERENCE INTEGRO-INTERPOLATION METHOD FOR DISCONTINUOUS SOLUTIONS OF THE USADEL EQUATIONS
Аннотация
The paper considers a one-dimensional problem for elliptic equations with nonstandard jump conditions on the inner boundary and a discontinuous solution. The integro-interpolation (balance) method is used to approximate the problem, including the junction condition on the inner boundary, which leads, in the case of Roben relations (the jump of the solution is proportional to the flux), to a four-point pattern. This difference scheme is used to solve the system of nonlinear Uzadel equations, which is the basic mathematical model at the microlevel for describing currents and fields in superconductors, including those with Josephson junctions. The results of calculations for the Abrikosov vortex problem are presented and the accuracy of the proposed approach is investigated, including for a simplified three-point scheme.
Differential equations. 2025;61(7):1000–1008
1000–1008