Vol 64, No 5 (2024)

This note serves as a supplement to a previous article by the author on the same topic. Its purpose is to more precisely characterize pairs of unitoids that allow simultaneous diagonalization.

The paper considers the problem of optimal distributed control in a strictly convex planar domain with a smooth boundary and a small parameter in one of the higher derivatives of the elliptic operator. In this problem, a zero Dirichlet boundary condition is imposed, and the control enters additively into the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square-integrable functions. The solutions to the resulting boundary value problems are treated in the generalized sense as elements of a certain Hilbert space. The optimality criterion is the sum of the square of the norm of the state deviation from a given state and the square of the norm of the control, with a weighting coefficient. This structure of the optimality criterion allows either the first or the second term to be emphasized, depending on the need. In the first case, achieving the desired state is prioritized, while in the second case, minimizing resource costs becomes more important. The asymptotics of the problem are studied in detail, arising from a second-order differential operator with a small coefficient in one of the higher derivatives, to which a zero-order differential operator is added.

The paper proposes a new method for numerically solving nonlinear stiff problems based on the numerical implementation of the holomorphic regularization method of the Cauchy problem for singularly perturbed nonlinear differential equations.

A class of exact solutions to the Navier-Stokes equations for axisymmetric vortex flows of incompressible fluids is obtained. Invariant manifolds of flows with rotational symmetry relative to a given axis in three-dimensional coordinate space are identified, and the structure of the solutions is described. It is established that typical invariant regions of such flows are rotational figures homeomorphic to a torus, forming a structure of topological fibration, such as in a sphere, cylinder, and more complex configurations composed of such figures. The results are extended to similar solutions of the magnetohydrodynamics (MHD) system and Maxwell’s electrodynamics equations, which possess ℝ3 analogous properties. Examples of axisymmetric vortex vector fields and the topological fibrations they generate on manifolds invariant ℝ3 under the dynamical systems defined by these fields are provided.

The most convenient model for describing the phenomenon of shock wave propagation in the atmosphere is the extended Burgers equation. This work investigates the influence of the numerical scheme on the results of solving the equation, which accounts for the nonlinear nature of shock wave propagation in the atmosphere. This equation is a key component of the extended Burgers equation and defines the transformation of the perturbed pressure profile during its propagation. Two numerical schemes were applied for the solution: CABARET and WENO, which are quasi-monotonic finite difference schemes that allow for solutions without significant numerical oscillations. An analysis of the applicability of these schemes for solving the considered problem was conducted.

It is generally accepted that the stabilizing correction scheme with central differences for spatial variables in the 3D transport equation is conditionally stable. The work shows that, strictly speaking, this scheme is absolutely unstable. However, the region of unstable harmonics in the wave vector space and the magnitude of their increments rapidly approach zero as the Courant parameter tends to zero, allowing successful use of this scheme. Therefore, it is more accurate to refer to this scheme as practically conditionally stable.