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卷 56, 编号 5 (2016)

Article

Combination of numerical and structured approaches to the construction of a second-order incomplete triangular factorization in parallel preconditioning methods

Milyukova O.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):699-716
pages 699-716 views

On exact estimates of the convergence rate of fourier series for functions of one variable in the space L2[–π, π]

Kerimov M., Selimkhanov E.

摘要

The work is devoted to exact estimates of the convergence rate of Fourier series in the trigonometric system in the space of square summable 2π-periodic functions with the Euclidean norm on certain classes of functions characterized by the generalized modulus of continuity. Some N-widths of these classes are calculated, and the residual term of one quadrature formula over equally spaced nodes for a definite integral connected with the issues under consideration is found.

Computational Mathematics and Mathematical Physics. 2016;56(5):717-729
pages 717-729 views

Two fast algorithms for projecting a point onto the canonical simplex

Malozemov V., Tamasyan G.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):730-743
pages 730-743 views

Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls

Kamenev G.

摘要

The estimate refinement method for the polyhedral approximation of convex compact bodies is analyzed. When applied to convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. In previous studies, for the approximation of a multidimensional ball, the convergence rates of the method were estimated in terms of the number of faces of all dimensions and the cardinality of the facial structure (the norm of the f-vector) of the constructed polytope was shown to have an optimal rate of growth. In this paper, the asymptotic convergence rate of the method with respect to faces of all dimensions is compared with the convergence rate of best approximation polytopes. Explicit expressions are obtained for the asymptotic efficiency, including the case of low dimensions. Theoretical estimates are compared with numerical results.

Computational Mathematics and Mathematical Physics. 2016;56(5):744-755
pages 744-755 views

Control of phase boundary evolution in metal solidification for new thermodynamic parameters of the metal

Albu A.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):756-763
pages 756-763 views

Implementation and efficiency analysis of an adaptive hp-finite element method for solving boundary value problems for the stationary reaction–diffusion equation

Zolotareva N., Nikolaev E.

摘要

An iterative process implementing an adaptive hp-version of the finite element method (FEM) previously proposed by the authors for the approximate solution of boundary value problems for the stationary reaction–diffusion equation is described. The method relies on piecewise polynomial basis functions and makes use of an adaptive strategy for constructing a sequence of finite-dimensional subspaces based on the computation of correction indicators. Singularly perturbed boundary value test problems with smooth and not very smooth solutions are used to analyze the efficiency of the method in the situation when an approximate solution has to be found with high accuracy. The convergence of the approximate solution to the exact one is investigated depending on the value of the small parameter multiplying the highest derivative, on the family of basis functions and the quadrature formulas used, and on the internal parameters of the method. The method is compared with an adaptive h-version of FEM that also relies on correction indicators and with its nonadaptive variant based on the bisection of grid intervals.

Computational Mathematics and Mathematical Physics. 2016;56(5):764-782
pages 764-782 views

Monotonicity of the CABARET scheme approximating a hyperbolic equation with a sign-changing characteristic field

Kovyrkina O., Ostapenko V.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):783-801
pages 783-801 views

Nonlocal unique solvability of a steady-state problem of complex heat transfer

Kovtanyuk A., Chebotarev A.

摘要

A boundary value problem of radiative–conductive–convective heat transfer in a threedimensional domain is proved to be uniquely solvable. An iterative algorithm is proposed for finding its solution.

Computational Mathematics and Mathematical Physics. 2016;56(5):802-809
pages 802-809 views

To the theory of volterra integral equations of the first kind with discontinuous kernels

Apartsin A.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):810-825
pages 810-825 views

Dynamics of chain particle aggregates in viscous flow

Martynov S., Tkach L.

摘要

The dynamics of aggregates consisting of chains of particles and their union in the form of a two-dimensional network in viscous flow is numerically simulated. It is assumed that the particles in a chain can move relative to each other so that the distance between two neighboring ones remains unchanged. The hydrodynamic interaction forces between all particles in an aggregate are taken into account. The deposition of particle chains and their dynamics in a linear flow are considered in the case an unbounded fluid volume and near a flat wall. The interaction forces between the particles necessary for retaining them in a chain are calculated, and places of the most probable breakage in the chain are determined.

Computational Mathematics and Mathematical Physics. 2016;56(5):826-840
pages 826-840 views

Detailed simulation of the pulsating detonation wave in the shock-attached frame

Lopato A., Utkin P.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):841-853
pages 841-853 views

Problem with nonequilibrium boundary conditions in the kinetic theory of gases

Aristov V., Zabelok S., Fedosov M., Frolova A.

摘要

The Boltzmann kinetic equation is considered in a new formulation with nonequilibrium distribution functions on free boundaries, which makes it possible to simulate nonequilibrium superand subsonic flows. Transport processes for such flows are analyzed. The possibility of anomalous transport is determined, in which case the heat flux, temperature gradient, and the corresponding components of the nonequilibrium stress tensor and the velocity gradient have the same sign.

Computational Mathematics and Mathematical Physics. 2016;56(5):854-863
pages 854-863 views

Discrete spectrum of cranked quantum and elastic waveguides

Nazarov S.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):864-880
pages 864-880 views

Effective averaging of stochastic radiative models based on Monte Carlo simulation

Ambos A., Mikhailov G.

摘要

Based on the Monte Carlo simulation and probabilistic analysis, stochastic radiative models are effectively averaged; that is, deterministic models that reproduce the mean probabilities of particle passage through a stochastic medium are constructed. For this purpose, special algorithms for the double randomization and conjugate walk methods are developed. For the numerical simulation of stochastic media, homogeneous isotropic Voronoi and Poisson mosaic models are used. The parameters of the averaged models are estimated based on the properties of the exponential distribution and the renewal theory.

Computational Mathematics and Mathematical Physics. 2016;56(5):881-893
pages 881-893 views

Resolving sequences of operators for linear ordinary differential and difference systems of arbitrary order

Petkovšek M., Ryabenko A., Abramov S.

摘要

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.

Computational Mathematics and Mathematical Physics. 2016;56(5):894-910
pages 894-910 views