Vol 9, No 3 (2017)

In solving stiff Cauchy problems, regions of a very rapid change in the solution, which are called boundary layers, arise. In nonlinear problems, there may be several such regions and not only at the initial moment but at different times when they are called contrast structures. It is shown that in the numerical solution of contrast structures, round-off errors can become so great that even a significant increase in the digit capacity is unable to resolve the situation. In this case, approximate analytical methods developed in boundary layer theory turn out to be more effective. As an applied problem, we consider a numerical solution of a real problem of chemical kinetics, i.e., combustion of hydrogen in oxygen. It is shown that this process involves the emergence of a contrasting structure due to the production of transitional chemical components. Therefore, the actual flare of the flame occurs not at the initial moment but after a delay.

A new method of automatic step selection is proposed for the numerical integration of the Cauchy problem for ordinary differential equations. The method is based on using the geometrical characteristics (cuvature and slope) of the integral curve. Formulas have been constructed for the curvature of the integral curve for different choices of multidimensional space. In the two-dimensional case, they turn into well-known formulas, but their general multidimensional form is nontrivial. These formulas have a simple form, are convenient for practical use, and are of independent interest for the differential geometry of multidimensional spaces. For the grids constructed by our method, a procedure of step splitting is proposed that allows one to apply Richardson’s method and to calculate posterior asymptotically precise error estimation for the obtained solution (no such estimates have been found for traditional algorithms of automatic step selection). Therefore, the proposed methods demonstrate significantly superior reliability and validity of the results as compared to calculations by conventional algorithms. In the existing automatic procedures for step selection, steps can be unexpectedly reduced by 2–4 orders of magnitude for no apparent reason. This undermines the reliability of the algorithms. The cause of this phenomenon is explained. The proposed methods are especially effective for highly stiff problems, which is illustrated by examples of calculations.

For a convex programming problem in a Hilbert space with operator equality constraints, the Lagrange principle in sequential nondifferential form or, in other words, the regularized Lagrange principle in iterative form, that is resistant to input data errors is proved. The possibility of its applicability for direct solving unstable inverse problems is discussed. As an example of such problem, we consider the problem of finding the normal solution of the Fredholm integral equation of the first kind. The results of the numerical calculations are shown.

An upper atmosphere probe is considered. For the atmosphere parameters, a reconstruction method using the density data by a vacuometer located behind a wire screen is constructed. In the context of the Monte-Carlo method of direct statistical simulation, we propose a simplified simulation algorithm for the interactions of a molecule with the wire screen treated as a semipermeable membrane. To obtain a relationship between the transmitter reading and the parameters of the undisturbed atmosphere, we numerically simulate the flow over the transmitter.

The statistical properties of dense random packings of ellipsoidal bodies in cylindrical vessels created under the effect of the gravity force are investigated by the numerical method developed from the equations of mechanics of a rigid body. The body-to-body and body-to-wall interaction is described by elastic forces. It has been found that the influence of the gravity force in the presence of walls specifies the statistically meaningful anisotropy of the orientation of ellipsoids. According to the method of the formation of a packing and the spatial separation of bodies by size can be found; this causes a local narrowing of the size spectrum. The method is naturally generalized by including friction forces, arbitrary force fields, and walls of an arbitrary shape.

We have obtained continuous analytical expressions approximating the Fermi-Dirac (F-D) integrals of orders j = −1/2, 1/2, 1, 3/2, 2, 5/2, 3, and 7/2 in a convenient form for calculation with reasonable accuracy (1–4)% in a wide degeneration range in this paper. An approach based on the least squares method for approximation was used. The demands for the approximation of integrals, the range of variation of order j, and the definitional domain are considered in terms of the use of F-D integrals to determine the properties of metals and semiconductors.