Vol 64, No 8 (2024)

A modification of the ROMB method is presented for numerically solving unsteady nonlinear heat conduction equations. The modified difference schemes are conservative and, for smooth solutions, approximate the model differential equations with constant coefficients with second-order accuracy in both time and space. First differential approximations are provided for the modified difference schemes, enabling the assessment of approximation errors in the schemes.

A model of competition between several microorganism species described by a system of nonlinear differential equations with infinite distributed delay is considered. The case of asymptotic stability of the equilibrium point corresponding to the survival of only one species and the extinction of all others is studied. The conditions for the initial numbers of species and the initial concentration of the nutrient at which the system reaches an equilibrium state are specified, and estimates of the stabilization rate are established. The results are obtained using the Lyapunov–Krasovskii functional.

The problem of estimating solutions and inverse matrices of systems of linear equations with a circulant matrix in the p-norm, 1 < p < w, is considered. An estimate is obtained for a circulant matrix with diagonal dominance. Based on this result and the idea of decomposing a matrix into a product of matrices related to the decomposition of the characteristic polynomial, an estimate is proposed for a general circulant matrix.

A new method for separating the matrix spectrum relative to the straight line is proposed, based on a fractional-linear transformation. It is noted that it has a number of advantages over approaches based on an exponential transformation, namely, its applicability area is wider, and the number of iterations required for its convergence is significantly smaller. The proposed method is used to study the problems of infinite strip flutter with various edge fixing conditions, which after suitable discretization of differential operators are reduced to spectral problems for matrices. The study of stability regions by the method of spectrum dichotomy relative to the imaginary axis allows one to construct neutral curves in the plane of parameters of the flutter problem.

The paper is devoted to the numerical analysis of the sensitivity of the characteristics of spatial stability of boundary layers to the errors with which the main flow is specified. It is proposed to use structured pseudospectra for this purpose. It is shown that the obtained estimates are significantly more accurate than the estimates based on the unstructured pseudospectrum. The presentation is carried out using the example of a viscous incompressible fluid flow over a concave surface of small curvature with flow parameters favorable for the development of Goertler vortices and Tollmien–Schlichting waves.

An inhomogeneous biharmonic equation is considered on the unit sphere in three-dimensional space. The solution of this equation, belonging to the Sobolev space on the sphere, is approximated by a sequence of solutions of the same equation but with specific right-hand sides, represented as linear combinations of shifts of the Dirac delta function. It is proven that, given specified nodes on the sphere determining the shifts, special solutions of the equation — spherical biharmonic splines — exist, and the weights corresponding to each are solutions of an associated non-degenerate system of linear algebraic equations. The connection between the approximation quality of the differential problem solution by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas is established.

A numerical modeling technique for unsteady flows of heat-conducting gas in a three-temperature approximation is presented. The methodology is based on the foundational principles of S.K. Godunov. For time integration, each time step is computed by splitting the governing equations into hyperbolic and parabolic subsystems. The first subsystem is solved using a generalized Godunov scheme, while the second uses an explicit-iterative Chebyshev scheme. Adaptive, curvilinear moving grids are used for discretization, and the discrete scheme is formulated in curvilinear coordinates, preserving the symmetries of the differential problem. The methodology is implemented as a parallel code for multiprocessor computers. Its primary purpose is to support computational studies in controlled thermonuclear fusion, though it can also be applied to other areas of computational aerogas dynamics.

This paper presents a new approach to numerical modeling of gas flow around stationary and moving rigid bodies, allowing for the use of Eulerian grids that are not tied to the geometry of the body. The bodies are assumed to be absolutely rigid and undeformable, with their elastic properties disregarded. The gas is inviscid and non-heat-conducting, described by compressible fluid equations. The proposed approach is based on averaging the equations of the original model over a small spatial filter. This results in a system of averaged equations that includes an additional quantity — the solid volume fraction parameter — whose spatial distribution digitally represents the geometry of the body (analogous to an order function). This system of equations operates across the entire space. Under this approach, the standard boundaryvalue problem within the gas region is effectively reduced to a Cauchy problem over the entire space. For a one-dimensional model, the numerical solution of the averaged equations is considered using Godunov’s method. In intersected cells, a discontinuous solution is introduced, leading to a compound Riemann problem that describes the decay of the initial discontinuity in the presence of a confining wall. It is shown that the approximation of the numerical flux for the compound Riemann problem solution ensures transport of the order function without numerical dissipation.