Rubric: "01.01.00 Mathematics"
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
Frolov A. V. (Water Problems Institute of the Russian Academy of Sciences)
doi: 10.18698/2309-3684-2016-3-7992
The paper considers long-term fluctuations of the Caspian Sea level as a nonlinear system output with positive and negative feedbacks. The Caspian Sea model with due consideration of an outflow into the Kara-Bogaz-Gol Bay is designed. Density distribution of the sea level is obtained as a solution to the corresponding Fokker — Planck — Kolmogorov equation. The bimodal probability density of the sea level distribution, which meets the endorheic Caspian Sea (if you cut off the Kara-Bogaz-Gol Bay), is shown to turn into the single-mode probability density in case of simultaneous influence of evaporation and seawater outflow into the Kara-Bogaz-Gol Bay on the sea level.
Frolov A. Modelling influence of outflow into the Kara-Bogaz-Gol Bay on probability density of the Caspian Sea level fluctuations. Маthematical Modeling and Coтputational Methods, 2016, №3 (11), pp. 79-92
519.6 Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems
Sulimov V. D. (Bauman Moscow State Technical University), Shkapov P. M. (Bauman Moscow State Technical University), Goncharov D. A. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-4-84102
The study examines extremum problems for eigen spectra components of Lagrangian dynamical systems. Mathematical models of the systems studied are described by the matrices depending on the parameters. The eigenproblems defined for such systems, in general, are characterized by a spectrum, which can contain multiple eigenvalues. Subtests in extremum problems are assumed to be continuous, Lipschitzian, multiextremum and maybe not everywhere differentiable functions. The search for global solutions is conducted using new hybrid algorithms that combine a stochastic algorithm for scanning the variables space and deterministic local search methods. The study gives numerical examples of solving the problems of global nondifferentiable minimization of the maximum systems eigenvalues.
Sulimov V., Shkapov P., Goncharov D. Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 84-102
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
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
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
Andrianov I. K. (Комсомольский-на-Амуре государственный технический университет), Grinkrug M. S. (Комсомольский-на-Амуре государственный технический университет)
doi: 10.18698/2309-3684-2016-2-2438
The paper considers a mathematical model of the heat exchange process occurring in the thin-walled curved shells of turbomachines. We propose an algorithm for calculating the thermal state on the boundary surfaces of the shell and of the coating according to the desired thermal condition. We present the calculation results of the temperature distribution in the given temperature field for the most terminally loaded shell surface caused by the heat effect.
Andrianov I., Grinkrug M. Parametric identification of a mathematical model of heat exchange process for thin-walled curved shells of turbomachines. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 24-38
Parkhomenko V. P. (ФИЦ ИУ РАН/Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-3-93109
The paper analyzes some factors affecting the parallel implementation performance of the atmospheric general circulation model designed on a cluster type multiprocessor computer. It considers several modifications of the initial parallel code of this model in order to improve both its computational efficiency and processor load balancing. The numerical scheme is modified according to the time of the atmospheric general circulation model for parallel computing of dynamics and physics blocks. The proposed procedure is used along with the procedures of paralleling the dynamics and physics blocks based on decomposition of the computational domain. It allows both optimizing the processor load balancing and increasing the paralleling efficiency. The data obtained while using the scheme for the physics block load balancing allow for complication of the physics block without increasing the total computational time. The results of numerical experiments are given.
Parkhomenko V. Algorithm for computational performance improvement and processor load balancing to simulate the general atmosphere circulation. Маthematical Modeling and Coтputational Methods, 2016, №3 (11), pp. 93-109
681.513.5 Stabilization of an unstable limit cycle of relay chaotic system
Krasnoschechenko V. I. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2015-2-87104
The article presents an algorithm of synthesis for stabilization of an unstable limit cycle of relay chaotic system. One-dimensional discrete Poincare map is used in algorithm for finding fixed points of the period one (limit cycles of initial continuous system). It is shown, that classical OGY method of dead beat regulator synthesis does not solve the problem as it takes into account only speed of the target coordinate what is not sufficient for stabilizing. The proposed algorithm is based on search of the necessary regulator factor by solving an inverse problem: at first some factor is assigned and then two-step procedure of system transition to the following switching point (with correction) is carried out. The task of correction is performed in a complete neighborhood of target coordinate position and speed, and it provides stabilization of a limit cycle by adjusting small amplitude pulses in the chosen area of entry conditions (area of stabilization) as evidenced by the simulation results.
Krasnoschechenko V. Stabilization of an unstable limit cycle of relay chaotic system. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 87-104