

2020 year, number 1
A.S. Kozelkov^{1,2}, S.V. Lashkin^{1}, A.A. Kurkin^{2}, A.V. Kornev^{3}, A.M. Vyalykh^{1}
^{1}Federal State Unitary Enterprise "Russian Federal Nuclear Center  AllRussian Research Institute of Experimental Physics", Nizhny Novgorod region, Sarov, Russia ^{2}Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia ^{3}Moscow Aviation Institute, Moscow, Russia
Keywords: вычислительная гидродинамика, алгоритм SIMPLE, многосеточный решатель, моделирование, computational fluid dynamics (CFD), SIMPLE algorithm, multigrid solver, modeling
Abstract >>
This paper deals with the investigation of parallel SIMPLE (SemiImplicit Method for Pressure Linked Equations) algorithm for the numerical solution of the NavierStokes system of equations for viscous incompressible flows. The interprocessor exchange mechanism with mesh decomposition with virtual cells and algebraic multigrid method is presented. The method of distributed matrix storage and the algorithm for matrixvector operations reducing the number of interprocessor exchanges are presented. The results of a series of the numerical experiments on structured and unstructured grids (including the external aerodynamics problem) are presented. Based on the results obtained, the analysis of the influence of multigrid solver settings on the total algorithm efficiency was made. It was shown that the parallel algorithm for the SIMPLE method based on the algebraic multigrid technique proposed makes possible to efficiently calculate problems on hundreds of processors.

C. Xu
School of Mathematics and Statistics, Beihua University, Jilin, China
Keywords: псевдогиперболические интегродифференциальные уравнения, задачи оптимального управления, априорные оценки ошибки, сверхсходимость, положительно определенные смешанные методы расщепления конечных элементов, pseudohyperbolic integrodifferential equations, optimal control problems, a priori error estimates, superconvergence, splitting positive definite mixed finite element methods
Abstract >>
In this paper, we discuss a priori error estimates and superconvergence of splitting positive definite mixed finite element methods for optimal control problems governed by pseudohyperbolic integrodifferential equations. The state variables and costate variables are approximated by the lowest order RaviartThomas mixed finite element functions, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates both for the control variable, the state variables and the costate variables. Second, we obtain a superconvergence result for the control variable.

A.I. Levykin^{1,2}, A.E. Novikov^{3}, E.A. Novikov^{3,4}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia ^{3}Siberian Federal University, Krasnoyarsk, Russia ^{4}Institute of Computational Mathematicsof the Federal Research Centerof the Krasnoyarsk Scientific Center,Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia
Keywords: методы типа Розенброка, дифференциальноалгебраические уравнения, жесткие системы ОДУ, Rosenbrocktype methods, differentialalgebraic equations, stiff systems of ODEs
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This paper deals with the derivation of the optimal form of the Rosenbrocktype methods in terms of the number of nonzero parameters and computational costs per step. A technique of obtaining ( m, k )methods from the wellknown Rosenbrocktype methods is justified. There are given formulas for the ( m, k )schemes parameters transformation for their two canonical representations and obtaining the form of a stability function. The authors have developed L stable (3, 2)method of order 3 which requires two evaluations of a function: one evaluation of the Jacobian matrix and one LU decomposition per step. Moreover, in this paper there is formulated an integration algorithm of the alternating step size based on (3, 2)method. It provides the numerical solution for both explicit and implicit systems of ODEs. The numerical results confirming the efficiency of the new algorithm are given.

K.V. Litvenko^{1}, S.M. Prigarin^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia
Keywords: simulation of random fields, spectral models, convergence, sea surface simulation, extreme ocean waves, rogue waves
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This paper deals with the convergence of spectral and conditional spectral models that are used to simulate a stochastic structure of the sea surface undulation and rogue ocean waves. We study the convergence of spatialtemporal and spatial models.

A.F. Mastryukov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Akad. Lavrentieva 6, Novosibirsk, 630090
Keywords: конечноразностный, оптимальный, точность, метод Лагерра, электромагнитные, система линейных уравнений, итерации, finite difference, optimal, accuracy, Laguerre method, electromagnetic, linear equations system, iterations
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This paper deals with a difference scheme of second order of approximation for onedimensional Maxwell's equations using the Laquerre transform. Supplementary parameters are introduced into this difference scheme. These parameters are obtained by minimizing the difference approximation error of the Helmholtz equation. The values of these optimal parameters are independent of the step size and the number of nodes in the difference scheme. It is shown that application of the Laguerre decomposition allows obtaining a higher accuracy of approximation of the equations in comparison with similar difference schemes when using the Fourier decomposition. The finite difference scheme of second order with parameters was compared to the difference scheme of fourth order in two cases. The use of an optimal difference scheme when solving the problem of electromagnetic impulse propagation in an inhomogeneous medium yields the accuracy of the solution compatible with that of the difference scheme of fourth order. When solving the inverse problem, the second order difference scheme makes possible to attain a higher accuracy of the solution as compared to the difference scheme of fourth order. In the considered problems, the application of the difference scheme of second order with supplementary parameters has decreased the calculation time of a problem by 2025 per cent as compared to the fourth order difference scheme.

R.K. Mohanty^{1}, S. Sharma^{2}
^{1}Department of Applied Mathematics, Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, India ^{2}Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India
Keywords: квазилинейные параболические уравнения, сплайн в напряжении, обобщенное уравнение БюргерсаХаксли, обобщенное уравнение БюргерсаФишера, итерационный метод Ньютона, quasilinear parabolic equations, spline in tension, generalized BurgersHuxley equation, generalized BurgersFisher equation, Newton's iterative method
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In this article, we discuss a fourthorder accurate scheme based on nonpolynomial spline in tension approximations for the solution of quasilinear parabolic partial differential equations. The proposed numerical method requires only two halfstep points and a central point on a uniform mesh in the spatial direction. This method is derived directly from a continuity condition of the firstorder derivative of a nonpolynomial tension spline function. The stability of the scheme is discussed using a model linear PDE. The method is directly applicable to solving singular parabolic problems in polar systems. The proposed method is tested on the generalized BurgersHuxley equation, the generalized BurgersFisher equation, and Burgers' equations in polar coordinates.

V.I. Paasonen^{1,2}
^{1}Institute of Computational Technologies SB RAS, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia
Keywords: симметричная разностная схема, компактная схема, симметричный шаблон, схема максимального порядка точности, многоточечная схема, многоточечный шаблон, symmetric difference scheme, compact scheme, symmetric stencil, scheme of maximal order of accuracy, multipoint scheme, multipoint stencil
Abstract >>
We study all possible symmetric twolevel difference schemes on arbitrary extended stencils for the Schrödinger equation and for the heat conductivity equation. We find the coefficients of the schemes from the conditions under which a maximum possible order of approximation on the main variable is attained. From a set of maximally exact schemes, a class of absolutely stable schemes is isolated. To investigate the stability of the schemes, the Neumann criterion is numerically and analytically verified. It is proved that the property of schemes to be absolutely stable or unstable significantly depends on the order of approximation on the evolution variable. As a result of the classification it was possible to construct absolutely stable schemes up to the tenth order of accuracy on the main variable.

