

2016 year, number 2
Galina Andreevna Babicheva^{1}, Nina Aleksandrovna Kargapolova^{1,2}, Vasilii Aleksandrovich Ogorodnikov^{1,2}
^{1}Novosibirsk State University, Pirogova st., 2, Novosibirsk, Russia, 630090 ^{2}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, Russia, 630090
Keywords: однородное случайное поле, стохастическое моделирование, рандомизация, homogeneous random field, stochastic simulation, randomization
Abstract >>
In this paper, two new algorithms for the simulation of homogeneous random fields are proposed. Both algorithms are based on the widespread algorithm “in rows and columns” for the simulation of the Gaussian fields with special correlation functions. Applying the algorithms developed makes possible to efficiently simulate homogeneous random fields with nonconvex correlation functions.

Vladimir Pavlovich Zhitnikov^{1}, Nataliya Mikhailovna Sherykhalina^{1}, Roza Ravilevna Muksimova^{2}
^{1}Ufa State Aviation Technical University, K. Marksa str., 12, Ufa, Russia, 450000 ^{2}SaintPetersburg State University of Civil Aviation, Pilotov str., 38, St. Petersburg, Russia, 196210
Keywords: уравнение теплопроводности, волновое уравнение, явная и неявная схемы, число Куранта, модели погрешности, численная фильтрация, heat equation, explicit and implicit schemes, the Courant number, model error, numerical filtration
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A mixed problem for a onedimensional heat equation with several versions of initial and boundary conditions is considered. Explicit and implicit schemes are applied for the solution. The sweep method and the iteration methods are used for the implicit scheme for solving the implicit system of equations. The numerical filtration of a finite sequence of results obtained for different grids with an increasing number of nodal points is used to analyze errors of the method and rounding. In addition, to investigate the rounding errors, the results obtained with several lengths of the machine word mantissa are compared. The numerical solution of the mixed problem for the wave equation is studied by similar methods. The occurrence of deterministic dependencies of the error in the numerical method and the rounding on spatial coordinates, time and the number of nodes is revealed. The source models to describe the behavior of errors in terms of time are based on the analysis of the results of numerical experiments for different versions of conditions of problems. In accord with such models, which were verified by the experiment, the errors can increase, decrease or stabilize depending on conditions over time similar to changing the energy or mass.

Sergey Igorevich Kabanikhin^{1,2}, Olga Igorevna Krivorotko^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090 ^{2}Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090
Keywords: уравнения мелкой воды, амплитуда фронта, фундаментальное решение, уравнение эйконала, конечноразностный метод, shallow water equations, tsunami amplitude, fundamental solution, eikonal equation, finite difference approach
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A numerical algorithm for computing tsunami wave front amplitude is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved by the Godunov approach and the bicharacteristic method. The qualitative comparison of the two above methods is described. Then a change in variables associated with the eikonal solution is introduced. At the last step, using the expansion of the fundamental solution of shallow water equations in the sum of singular and regular parts, we obtain the Cauchy problem for the wave amplitude. This approach allows one to reduce computer costs. The numerical results are presented.

Munish Kansal, V. Kanwar, Saurabh Bhatia
Panjab University, Chandigarh160 014, India
Keywords: области притяжения, метод Ньютона, методы Кинга, оптимальные итерационные методы, показатель эффективности, basins of attraction, Newton's method, King's methods, optimal iterative methods, efficiency index
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In this paper, we present new interesting fourthorder optimal families of ChebyshevHalley type methods free from secondorder derivatives. In terms of computational cost, each member of the families requires two functions and one firstorder derivative evaluation per iteration, so that their efficiency indices are 1.587. It is found by way of illustration that the proposed methods are useful in high precision computing environment. Moreover, it is also observed that larger basins of attraction belong to our methods, whereas the other methods are slow and have darker basins, while some of the methods are too sensitive to the choice of the initial guess.

Vladimir Dmitrievich Korneev^{1,2}, Viktor Mitrofanovich Sveshnikov^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090 ^{2}Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090
Keywords: краевые задачи, методы декомпозиции области, уравнение ПуанкареСтеклова, квазиструктурированные сетки, алгоритмы и технологии распараллеливания, boundary value problems, domain decomposition methods, PoincareSteklov equation, quasistructured grids, algorithms and technologies of parallelization
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A new approach to the decomposition method of a threedimensional computational domain into subdomains, adjoint without overlapping, which is based on a direct approximation of the PoincareSteklov equation at the conjugation interface, is proposed. With the use of this approach, parallel algorithms and technologies for threedimensional boundary value problems on quasistructured grids are presented. The experimental evaluation of the parallelization efficiency on the solution of the model problem on quasistructured parallelepipedal coordinated and uncoordinated grids is given.

Oleg Gennad'evich Monakhov^{1}, Emiliya Anatol'evna Monakhova^{1}, Millie Pant^{2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090 ^{2}New Technology Block, Saharanpur Campus of IIT, Roorkee, Saharanpur247667, India
Keywords: торговые стратегии, алгоритм дифференциальной эволюции, финансовый индикатор, эволюционные вычисления, trading strategy, parallel genetic algorithm, technical analysis, financial indicator, template, evolutionary computation
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An approach to optimization of trading strategies (algorithms) based on indicators of financial markets and evolutionary computation is described. A new version of the differential evolution algorithm for the search for optimal parameters of trading strategies for the trading profit maximization is used. The experimental results show that this approach can considerably improve the profitability of the trading strategies.

LarisaValentinovna Stepanova, Ekaterina Michailovna Yakovleva
Samara State University, Akad. Pavlov str., 1 Samara 443011
Keywords: нелинейная задача на собственные значения, метод возмущений, асимптотика напряжений и деформаций в окрестности вершины трещины, смешанное нагружение образца с трещиной, степенной определяющий закон, спектр собственных значений, nonlinear eigenvalue problem, perturbation theory small parameter method, asymptotics of stress and strain fields in the vicinity of the mixedmode crack, mixedmode loading, power constitutive law, eigenspectrum
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In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in the nonlinear fracture mechanics in analysis of stressstrain fields near a crack tip under a mixed mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the small artificial parameter method). The artificial small parameter is a difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique gives an effective method of solving nonlinear eigenvalue problems in the nonlinear fracture mechanics. Comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising from the fracture mechanics analysis in the case of a mixed mode loading.

Alena Victorovna Favorskaya, Igor Borisovich Petrov
Moscow Institute of Physics and Technology, Inststitutskii per., 9, Dolgoprudny, Moscow region, Russia, 141700
Keywords: сеточнохарактеристический метод, численное моделирование, неструктурированные сетки, интерполяция высоких порядков, gridcharacteristic method, numerical simulation, unstructured grids, high order interpolation
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We study the gridcharacteristic methods for solving hyperbolic systems using a high order interpolation on unstructured tetrahedral and triangular grids for approximation. We consider the interpolation with orders from the first to the fifth included. Also, onedimensional finite difference schemes appropriate for the considered methods are given. We study these schemes in terms of stability. The gridcharacteristic method on unstructured triangular and tetrahedral grids are successfully used for solving the seismic prospecting problems, including, seismic prospecting in the conditions of the Arctic shelf and permafrost, as well as for solving seismic problems, problems of dynamic deformation and destruction, studying anisotropic composite materials.

