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Siberian Journal of Numerical Mathematics

2019 year, number 3

1.
A modification of numerical methods for stochastic differential equations with the first integral

T.A. Averina1,2, K.A. Rybakov3
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
3Moscow Aviation Institute, Moscow, Russia
Keywords: численные методы, статистическое моделирование, стохастические дифференциальные уравнения, многообразие, первый интеграл, проекция, numerical methods, statistical modeling, stochastic differential equations, manifold, first integral, projection

Abstract >>
In this paper, stochastic differential equations (SDEs) with the first integral are considered. The exact solution of such SDEs belongs to a smooth manifold with probability 1. However, the numerical solution does not belong to the manifold, but it belongs to some of its neighborhood due to the numerical error. The main objective of the paper is to construct modified numerical methods for solving SDEs that preserve the first integral. In this study, exact solutions for three SDE systems with the first integral are obtained, and the proposed modification of numerical methods is tested on these systems.
																								



2.
Simulation of body motion in viscous incompressible fluid

A.S. Kozelkov1,2, V.R. Efremov3, A.A. Kurkin2, N.V. Tarasova1, D.A. Utkin1, E.S. Tyatyushkina2
1Federal State Unitary Enterprise, Sarov, Russia
2Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia
3JSC В«KBP named after Academician A. Shipunov», Tula, Russia
Keywords: технология В«химераВ», алгоритм SIMPLE, неструктурированные сетки, многофазность, твердое тело, chimera methodology, SIMPLE method, unstructured grid, multiphase, rigid body

Abstract >>
The description of a method for modeling the motion of bodies in a viscous incompressible fluid with the use of counting technology on grids with overlapping («chimera» technology) is given. Equations describing the flow of a viscous incompressible fluid are approximated by the finite volume method on an arbitrary unstructured grid. Their iterative solution is implemented using the SIMPLE algorithm. The description of the basic equations in the case of grid motion is given in this paper. The features of realizing the conditions on the boundaries of the grid regions that are established during the construction of the interpolation template are described. A method for overcoming numerical instability in the use of a rigid body model is demonstrated. The feature of taking into account the forces of gravitation in the case of the presence of multiphase media is described. The results of solving the problem of the motion of a cylinder in a fluid, the problem of the drop of a sphere into a fluid, and the problem of the ship's model flooding are presented.
																								



3.
Exact solutions of shallow water equations for the water oscillation problem in a simulated basin and their implementation in verifying numerical algorithms

N.A. Matckevich1,2, L.B. Chubarov1,2
1Novosibirsk State University, Novosibirsk, Russia
2Institute of Computational Technologies SB RAS, Novosibirsk, Russia
Keywords: накат волн на берег, свободная поверхность, сила Кориолиса, сила донного трения, математическое моделирование, уравнения мелкой воды, аналитические решения, обыкновенные дифференциальные уравнения, численные алгоритмы, метод крупных частиц, верификация, wave run-up, free surface, Coriolis force, bottom friction, mathematical modeling, shallow water equations, exact solutions, ordinary differential equations, numerical algorithms, large particles method, verification

Abstract >>
We present the approaches to solving a problem of shallow water oscillations in a parabolic basin (including an extra case of a horizontal plane). A series of assumptions about the form of solution and effects of Earth`s rotation and bottom friction are made. Then the resulting ODE systems are solved. The corresponding free surfaces have first or second order. The conditions of finiteness and localization of a flow are analyzed. The solutions are used in the verification of numerical algorithm of the large particles method, the efficiency of the carried out tests is discussed.
																								



4.
Numerical methods for a nonlocal parabolic problem with nonlinearity of Kirchhoff type

M. Mbehou, G. Chendjou
University of Yaounde I, Yaounde, Cameroon
Keywords: Оё-схема, уравнение Кирхгофа, нелокальный член диффузии, оптимальная оценка ошибки, метод конечных элементов Галеркина, Оё-scheme, Kirchhoff equation, nonlocal diffusion term, optimal error estimate, Galerkin finite element method

Abstract >>
The presence of the nonlocal term in the nonlocal problems destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, this paper is devoted to the analysis of a linearized Theta-Galerkin finite element method for the time-dependent nonlocal problem with nonlinearity of Kirchhoff type. Hereby, we focus on time discretization based on θ -time stepping scheme with θ ∈ [1/2,1). Some a error estimates are derived for the standard Crank-Nicolson ( θ =1/2), the shifted Crank-Nicolson ( θ = 1/2 + δ, δ is the time-step) and the general case ( θ ≠ 1/2 + , k = 0,1). Finally, numerical simulations that validate the theoretical findings are exhibited.
																								



5.
On the controllability of one new non-parametric statistical criterion, alternative to the Wilcoxon-Mann-Whitney criterion

G.I. Salov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: две выборки, непараметрический критерий, управляемый непараметрический критерий, two-sample problem, controllable non-parametric statistical test

Abstract >>
In this paper, we introduce the notion of controllability of a non-parametric statistical test and compare the powers of one new controllable non-parametric statistical test and the Wilcoxon-Mann-Whitney test for the cases with samples from exponential distribution.
																								



6.
Mathematical modeling of positive connection functioning in the tumor markers p53-microRNA system

S.D. Senotrusova, O.F. Voropaeva
Institute of Computational Technologies SB RAS, Novosibirsk, Russia
Keywords: математическое моделирование, уравнение с запаздыванием, онкомаркер, р53, Mdm2, микроРНК, положительная обратная связь, miR-34, miR-145, рак, mathematical modeling, delay equation, tumor marker, p53, Mdm2, microRNA, positive feedback, miR-34, miR-145, cancer

Abstract >>
The hierarchy of minimal mathematical models of the dynamics of p53-Mdm2-microRNA system has been developed. The models are based on the differential equations with time delay, hiding complex mechanisms of interaction in the signal system of the p53 protein. We consider the two types of interaction of p53 with microRNAs: the positive direct connection and the positive feedback. The feedback of microRNA-p53 is due to the negative effect of microRNA on the protein Mdm2, which itself is a negative regulator of p53. To approximate the direct positive effect of p53 on the microRNA, a linear function or the Goldbeter-Koshland type representation is used. The comparison of numerical solutions with medical and biological data of a number of specific p53-dependent microRNAs is made, which proves the adequacy of the models proposed and the results of numerical analysis. Special attention was given to the analysis of the positive feedback of p53 and microRNAs. The minimal models adopted have allowed us to consider the most general regularities of the p53-dependent microRNAs functioning. In the framework of these mathematical models it is shown that it is possible to neglect the connection Mdm2-miRNA for, at least, some of the most studied microRNAs associated with a direct positive connection with p53. However, those of the microRNAs, which are important negative regulator Mdm2, can have the most significant impact on the functioning of the entire system p53-Mdm2-microRNA. Conditions were obtained under which the regulatory function of microRNAs with respect to p53 is manifested. The results of numerical experiments indicate that such microRNAs can be considered to be a factor of the anticancer therapy.
																								



7.
Asymptotic analysis of the crack tip stress field (consideration of higher order terms)

L.V. Stepanova
Samara State University, Samara, Russia
Keywords: напряженно-деформированное состояние у вершины трещины, многопараметрическое описание поля напряжений у вершины трещины, смешанное деформирование, коэффициент интенсивности напряжений, T-напряжения, коэффициенты высших приближений, stress-strain state near the crack tip, multi-parameter asymptotic description of the stress field, mixed-mode loading, stress intensity factor, T-stress, coefficients of higher order terms

Abstract >>
This paper deals with the multi-parameter asymptotic description of the stress field near the crack tip of a finite crack in an infinite isotropic elastic plane medium subject to 1) tensile stress; 2) in-plane shear; 3) mixed mode loading for a wide range of mode-mixing situations (Mode I and Mode II). The multi-parameter series expansion of the stress tensor components containing higher order terms has been constructed. All the coefficients of the multi-parameter series expansion of the stress field are given. The main focus is on the discussion of the influence of considering the higher-order terms of the Williams expansion. Analysis of the higher order terms in the stress field is made. It is shown that the larger distance from the crack tip, the more terms are necessary to be kept in the asymptotic series expansion. Therefore, it can be concluded that several more higher-order terms of the Williams expansion must be used for the stress field description when the distance from the crack tip is not small enough. The crack propagation direction angle has been calculated. Two fracture criteria: maximum tangential stress criterion and the strain energy density criterion, are used. The multi-parameter form of two commonly used fracture criteria is introduced and tested. Thirty and more terms of the Williams expansion enable the angle to be calculated more precisely.
																								



8.
Variational interpolation of functionals in transport theory inverse problems

V.V. Uchaikin1, V.A. Litvinov2
1Ulyanovsk State University, Ulyanovsk, Russia
2Barnaul law Institute of MIA of Russia, Barnaul, Russia
Keywords: теория возмущений, сопряжённые функции, операторы, стационарный функционал, perturbation theory, adjoint functions, operators, stationary functional

Abstract >>
It is known that the dual representation of problems (through solutions of the main and the conjugate in the Lagrange sense equations) allows one to formulate the perturbation theory serving as basement for the successive approximation method in the inverse problems theory. If, according to preliminary predictions, the solution of an inverse problem (for example, the structure of the medium of interest) belongs to a certain set A, then selecting a suitable (trial, reference) element a 0 as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of the solution of the direct problem in this domain and find a subset A0 that best matches the measurement data. However, as the accuracy requirements increase, the domain A0 of the first approximation is rapidly narrowing, expanding it by adding higher terms of the expansion complicates the decision procedure. For this reason, a number of works have been devoted to the search for unperturbed approaches. Among them is the method of variational interpolation (VI-method), in the frame of which not one, but several problems a1, a2, …, an are used in order to compose from their solutions the desired one. The functional of interest is represented in the stationary form, and the coefficients of the expansion are determined from the condition of stationarity of the bilinear form. This paper demonstrates the application of VI-method to solving inverse problems in the frame of simple model situations associated with cosmic rays astrophysics.