Том 58, № 10 (2018)

Problems for time-dependent equations in which processes are characterized by different time scales are studied. Parts of the equations describing fast and slow processes are distinguished. The basic features of such problems related to their approximation are taken into account using finer time grids for fast processes. The construction and analysis of inhomogeneous time approximations is based on the theory of additive operator–difference schemes (splitting schemes). To solve time-dependent problems with different time scales, componentwise splitting schemes and vector additive schemes are used. The capabilities of the proposed schemes are illustrated by numerical examples for the time-dependent convection–diffusion problem. If convection is dominant, the convective transfer is computed on a finer time grid.

Modifications of Godunov’s scheme based on V.P. Kolgan’s approach to the construction of second-order accurate schemes in space for smooth solutions are proposed. The gasdynamic parameters linearly interpolated in a mesh cell, but the Riemann problem is solved for parameters at an intermediate point between the center and the boundary of the cell. Properties of Kolgan’s scheme and the proposed modifications, such as monotonicity and entropy nondecrease, are examined as applied to the system of differential equations describing plane sound waves propagation in a resting gas. Test problems in nonlinear gas dynamics are solved, namely, the Riemann problem in a pipe, transformation of an inhomogeneity in a plane-parallel flow, supersonic flow entering an axisymmetric convergent-divergent nozzle with a coaxial central body, and coaxial supersonic flow around a cylinder. The efficiency of schemes with an intermediate point is demonstrated.

A nonlinear Sturm–Liouville-type eigenvalue problem on an interval with a boundary condition of the first kind and an additional local condition at one of the boundaries of the interval is considered. All the parameters of the problem are real. The existence of an infinite number of (isolated) positive eigenvalues is proven, their asymptotic behavior is indicated, a condition for the periodicity of the eigenfunctions is found, the period is calculated, and an explicit formula for the zeros of the eigenfunction is presented. It is shown that methods of perturbation theory are not applicable to the complete study of the nonlinear problem.

An indirect boundary element method is proposed for the numerical solution of the Stokes equations in the axisymmetric case with Dirichlet and Neumann boundary conditions, which correspond to the velocity given on a part of the boundary and to the traction given on the other part. The method is described as applied to quasi-steady viscous creeping flows under gravity in regions bounded by a free surface and solid walls. A viscous fluid drop spreading over a hydrophobic horizontal surface is considered as an example. It is shown that the use of no-slip conditions ensure that the results have approximation convergence and satisfy the mass and energy conservation laws.

A new Godunov-type scheme for a complete nonlinear system of equations is developed as applied to the simulation of flows in an elastic deformable pipe. The proposed finite-difference method makes use of a one-dimensional grid, but the grid cells are characterized by two spatial sizes. The multidimensional effects and, primarily, pipe deformations are taken into account in the course of grid cell reconstruction. Difference relations are written for such a reconstructed grid. Due to the grid reconstruction procedure, the elastic deformation of the pipe can be taken into account directly without introducing the velocity of propagation of perturbations in the system “weakly compressible fluid–cylindrical shell.” The algorithm was shown to be highly accurate as applied to the water hammer problem.

Conditions under which it is possible to design a correct algorithm and a six-level spatial neural network reproducing the computations performed by this algorithm for recognition problems with binary data (Ω-regular problems) are found. A distinctive feature of this network is the use of diagonal activation functions in its internal layers, which significantly simplify intermediate computations in the inner and outer loops. Given an Ω-regular problem, the network sequentially computes the rows of the classification matrix for the test sample objects. The computational process (i.e., the inner loop) for each test object consists inside the elementary 3-level network (i.e., μ-block) of a single iteration determined by a single object of the training set. The proposed approach to the neural network construction does not rely on the conventional approach based on the minimization of a functional; rather, it is based on the operator theory developed by Zhuravlev for solving recognition and classification problems.