Volume 56, Nº 5 (2016)

Parallel versions of the stabilized second-order incomplete triangular factorization conjugate gradient method in which the reordering of the coefficient matrix corresponding to the ordering based on splitting into subdomains with separators are considered. The incomplete triangular factorization is organized using the truncation of fill-in “by value” at internal nodes of subdomains, and “by value” and ‘by positions” on the separators. This approach is generalized for the case of constructing a parallel version of preconditioning the second-order incomplete LU factorization for nonsymmetric diagonally dominant matrices with. The reliability and convergence rate of the proposed parallel methods is analyzed. The proposed algorithms are implemented using MPI, results of solving benchmark problems with matrices from the collection of the University of Florida are presented.

Two fast orthogonal projection algorithms of a point onto the canonical simplex are analyzed. These algorithms are called the vector and scalar algorithms, respectively. The ideas underlying these algorithms are well known. Improved descriptions of both algorithms are given, their finite convergence is proved, and exact estimates of the number of arithmetic operations needed for their implementation are derived, and numerical results of the comparison of their computational complexity are presented. It is shown that on some examples the complexity of the scalar algorithm is maximal but the complexity of the vector algorithm is minimal and conversely. The orthogonal projection of a point onto the solid simplex is also considered.

The problem of controlling the phase boundary evolution in the course of solidification of metals with different thermodynamic properties is studied. The underlying mathematical model of the process is based on a three-dimensional nonstationary two-phase initial–boundary value problem of the Stefan type. The control functions are determined by optimal control problems, which are solved numerically with the help of gradient optimization methods. The gradient of the cost function is exactly computed by applying the fast automatic differentiation technique. The research results are described and analyzed. Some of them are illustrated.

The monotonicity of the CABARET scheme approximating a hyperbolic differential equation with a sign-changing characteristic field is analyzed. Monotonicity conditions for this scheme are obtained in domains where the characteristics have a sign-definite propagation velocity and near sonic lines, on which the propagation velocity changes its sign. These properties of the CABARET scheme are illustrated by test computations.

A nonclassical Volterra linear integral equation of the first kind describing the dynamics of an developing system with allowance for its age structure is considered. The connection of this equation with the classical Volterra linear integral equation of the first kind with a piecewise-smooth kernel is studied. For solving such equations, the quadrature method is applied.

The paper is devoted to the numerical investigation of the stability of propagation of pulsating gas detonation waves. For various values of the mixture activation energy, detailed propagation patterns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of computational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm.

The spectrum of quantum and elastic waveguides in the form of a cranked strip is studied. In the Dirichlet spectral problem for the Laplacian (quantum waveguide), in addition to well-known results on the existence of isolated eigenvalues for any angle α at the corner, a priori lower bounds are established for these eigenvalues. It is explained why methods developed in the scalar case are frequently inapplicable to vector problems. For an elastic isotropic waveguide with a clamped boundary, the discrete spectrum is proved to be nonempty only for small or close-to-π angles α. The asymptotics of some eigenvalues are constructed. Elastic waveguides of other shapes are discussed.

We introduce the notion of a resolving sequence of (scalar) operators for a given differential or difference system with coefficients in some differential or difference field K. We propose an algorithm to construct, such a sequence, and give some examples of the use of this sequence as a suitable auxiliary tool for finding certain kinds of solutions of differential and difference systems of arbitrary order. Some experiments with our implementation of the algorithm are reported.