Rubric: "01.01.00 Mathematics"
519.612.2 Performance analysis of iterative methods of combined linear algebraic equations solution
Marchevsky I. K. (Bauman Moscow State Technical University), Puzikova V. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2014-4-3752
When sampling partial differential equations one has to solve a system of linear algebraic equations. To select the optimal in the sense of the computational efficiency of iterative method for solving such equations, in addition to the rate of convergence we should take into account such characteristics of the system and method, as the condition number, the smoothing factor, the indicator "costs on." The last two characteristics are calculated by the coefficients of harmonics amplification that give evidence of the smoothing properties of the iterative method and its "costs on", i. e. how worse the method suppresses frequency components of the error as compared with the highfrequency ones. The suggested method of determining harmonic gain factors is based on of the discrete Fourier transform. As an example, an analysis of the effectiveness of the BiCGStab method with ILU and multigrid preconditioning when solving difference analogues of the Helmholtz and Poisson equations is described.
Marchevsky I., Puzikova V. Performance analysis of iterative methods of combined linear algebraic equations solution. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 37-52
519.63 Parallel multigrid algorithms
Martynenko S. I. (Baranov Central Institute of Aviation Motor Development)
doi: 10.18698/2309-3684-2015-2-105120
The paper represents the main directions of development of the parallel classic multigrid algorithms and discusses their disadvantages. The possibility of efficient parallelization of smoothing iterations at the levels of coarse grids is shown using the Robust Multigrid Technique. Then multigrid structure is used for developing hybrid multigrid method. The paper describes estimations of speed-up and efficiency of different parallel multigrid algorithms as well as the results of numerical experiments.
Martynenko S. Parallel multigrid algorithms. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 105-120
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics), Sorokin V. G. (Ishlinsky Institute for Problems in Mechanics), Vyazmin A. V. (Moscow State University of Mechanical Engineering)
doi: 10.18698/2309-3684-2014-4-5373
In the article we explored nonlinear hyperbolic delay reaction-diffusion equations with varying transfer coefficients. A number of generalized separable solutions were obtained. Most of the equations considered contain arbitrary functions. Global nonlinear instability conditions of solutions of hyperbolic delay reaction-diffusion systems were determined. The generalized Stokes problem for a linear delay diffusion equation with periodic boundary conditions was solved.
Polyanin A., Sorokin V., Vyazmin A. Nonlinear delay reaction-diffusion equations of hyperbolic type: Exact solutions and global instability. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 53-73
Dimitrienko Y. I. (Bauman Moscow State Technical University), Dimitrienko O. Y. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-1-105122
The article considers the concept of applying the multidimensional continuum model to one of the main problems emerging in the theory of large scale data array treatment i.e. forecasting the dynamics of data cluster change. The concept is based on the model of multidimensional continua in spaces of high dimensionality (more than three) earlier developed by the authors. The model includes the integral conservation laws, which are reformulated for informational data clusters, as well as the model of motion kinematics and cluster deformation. The model of deformable multidimensional cluster is developed. The movement of the cluster in multidimensional data space includes translational and rotational motion and uniform tension-compression strain. The system of differential tensor equations describing the dynamics of the deformable multivariate cluster motion over time is formulated. A numerical algorithm for solving the system of differential equations for the ellipsoidal model of multidimensional cluster is worked out. An example of the developed model application for predicting the dynamics of economic data (data on goods purchases in a large supermarket) is considered. The results of forecasting the data on purchases of different consumer groups are shown.
Dimitrienko Y., Dimitrienko O. A model of multidimensional deformable continuum for forecasting the dynamics of large scale array of individual data. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 105-122
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics), Zhurov A. I. (Cardiff University/Ishlinsky Institute for Problems in Mechanics)
doi: 10.18698/2309-3684-2015-4-337
We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x F(u,w),where w = u(x,t) and w = u(x,t ), with denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N
n n n u x t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.
Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37
519.8 Stochastic models of the two unit duel fight
Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-2-6984
On the basis of the theory of continuous Markov processes we developed models of the two unit duel fight. We obtained computing formulas for calculating the basic fight indicators. Moreover, we found that the pre-emptive strike of one of the units participating in the fight has a significant impact on the fight outcome of the units which are similar in forces. The strike has a negligible impact, if one of the units has a significant advantage. The findings of the research show that the use of model with constant effective firing rates can lead to significant errors in the evaluation of its results. Finally, we found that the pre-emptive strike, coupled with a high degree of effective firing rate growth, can sometimes compensate for more than the double initial superiority of the opponent. We show the possibility of using approximations of the effective firing rate of the fighting units by the different functions of the fight time.
Chuev V., Dubogray I. Stochastic models of the two unit duel fight. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 69-84
Volkov V. Y. (JSC OKB GIROPRESS,), Golibrodo L. A. (JSC OKB GIROPRESS,), Zorina I. G. (Bauman Moscow State Technical University), Kudryavtsev O. V. (JSC OKB GIROPRESS,), Krutikov A. A. (JSC OKB GIROPRESS,), Skibin A. A. (JSC OKB GIROPRESS,)
doi: 10.18698/2309-3684-2016-4-3446
For modeling piping systems we made a transition from the mass balance equations, based on 1m and 2m Kirchhoff laws, to the mathematical description of a hydraulic network using the continuity equation discretization. For this purpose we applied a controlvolume method. This paper introduces an extension of the developed control-volume method for extended period simulations in hydraulic networks. This extension is developed for slow time-varying conditions in the hydraulic networks and is not intended to calculate rapidly occurring local phenomena such as waterhammer. The control-volume method was successfully applied to test tasks.
Volkov V., Golibrodo L., Zorina I., Kudryavtsev O., Krutikov A., Skibin A. Applying the control−volume method to extended period simulations in pipe network hydraulics. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 34-46
519.63 Development and testing for methods of solving stiff ordinary differential equations
Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Khodzhaeva S. R. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2014-4-95119
The paper is aimed at research of the (m,k)-method, CROS, finite superelement method and 4-stage explicit Runge–Kutta method for solving stiff systems of ordinary differential equations. Analysis of tests results showed that the best choice for systems with high stiffness is CROS. The finite superelement method is the «precise» method for solving linear systems of ordinary differential equations, the best supporting method for its implementation is (4,2)-method. The variation of the finite superelement method has been built and tested for solving nonlinear problems, this method proved to be unsuitable for problems with high stiffness.
Galanin M., Khodzhaeva S. Development and testing for methods of solving stiff ordinary differential equations. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 95-119
Rodnikov A. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-3-110118
We suggest a numerical and analytical algorithm of searching for stationary space station orbits in the vicinity of an oblate asteroid, when these orbits correspond to relative equilibrium positions of the space station on the plane defined by the asteroid precession and proper rotation axes, in the case of the asteroid being represented by a solid body of an approximately dynamically symmetric shape, compressed along the axis of dynamic symmetry. The algorithm is based on representing the asteroid gravitational potential by a composition of two complex conjugate point masses and consists of sequential variable substitution steps, reducing the problem to solving algebraic equations analytically and numerically. We supply a number of facts concerning evolution of stationary orbits in cases of changes in precession angular velocity.
Rodnikov A. Modelling the search for stationary space station orbits in the vicinity of an oblate-shaped asteroid. Маthematical Modeling and Coтputational Methods, 2016, №3 (11), pp. 110-118