Volume 56, Nº 2 (2016)

In a finite-dimensional linear space, consider a nonlinear eigenvalue problem analytic with respect to its spectral parameter. The notion of a principal vector for such a problem is examined. For a linear eigenvalue problem, this notion is identical to the conventional definition of principal vectors. It is proved that the maximum number of linearly independent eigenvectors combined with principal (associated) vectors in the corresponding chains is equal to the multiplicity of an eigenvalue. A numerical method for constructing such chains is given.

The linear regression problem is considered as an improper interpolation problem. The metric l₁ is used to correct (approximate) all the initial data. A probabilistic justification of this metric in the case of the exponential noise distribution is given. The original improper interpolation problem is reduced to a set of a finite number of linear programming problems. The corresponding computational algorithms are implemented in MATLAB.

Numerical methods for solving optimal control problems with equality constraints at the right end of the trajectory are discussed. Algorithms for optimal control search are proposed that are based on the multimethod technique for finding an approximate solution of prescribed accuracy that satisfies terminal conditions. High accuracy is achieved by applying a second-order method analogous to Newton’s method or Bellman’s quasilinearization method. In the solution of problems with direct control constraints, the variation of the control is computed using a finite-dimensional approximation of an auxiliary problem, which is solved by applying linear programming methods.

The behavior of the formal solution, obtained by the Fourier method, to a mixed problem for the wave equation with arbitrary two-point boundary conditions and the initial condition φ(х) (for zero initial velocity) with weaker requirements than those for the classical solution is analyzed. An approach based on the Cauchy–Poincare technique, consisting in the contour integration of the resolvent of the operator generated by the corresponding spectral problem, is used. Conditions giving the solution to the mixed problem when the wave equation is satisfied only almost everywhere are found. When φ(x) is an arbitrary function from L₂[0, 1], the formal solution converges almost everywhere and is a generalized solution to the mixed problem.

The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form φ(u) = u⁴–u² is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.

Issues concerning the implementation and practical application of geometric and algebraic multigrid techniques for solving systems of difference equations generated by the finite volume discretization of the Euler and Navier–Stokes equations on unstructured grids are studied. The construction of prolongation and interpolation operators, as well as grid levels of various resolutions, is discussed. The results of the application of geometric and algebraic multigrid techniques for the simulation of inviscid and viscous compressible fluid flows over an airfoil are compared. Numerical results show that geometric methods ensure faster convergence and weakly depend on the method parameters, while the efficiency of algebraic methods considerably depends on the input parameters.

Plane and axisymmetric cavitation flow problems are considered using Riabouchinsky’s scheme. The incoming flow is assumed to be irrotational and steady, and the fluid is assumed to be inviscid and incompressible. The flow problems are solved by applying the boundary element method with quadrature formulas without saturation. The free boundary is determined using a gradient descent technique based on Riabouchinsky’s principle. The drag force acting on the cavitator is expressed in terms of the Riabouchinsky functional. As a result, for small cavitation numbers, the force is calculated with a fairly high accuracy. Dependences of the drag coefficient are investigated for variously shaped cavitators: a wedge, a cone, a circular arc, and a spherical segment.