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Siberian Journal of Numerical Mathematics

2020 year, number 1

Effective implementation of the parallel SIMPLE algorithm based on multigrid method

A.S. Kozelkov1,2, S.V. Lashkin1, A.A. Kurkin2, A.V. Kornev3, A.M. Vyalykh1
1Federal State Unitary Enterprise "Russian Federal Nuclear Center - All-Russian Research Institute of Experimental Physics", Nizhny Novgorod region, Sarov, Russia
2Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia
3Moscow 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 (Semi-Implicit Method for Pressure Linked Equations) algorithm for the numerical solution of the Navier-Stokes 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 matrix-vector 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.

A priori error estimates and superconvergence of splitting positive definite mixed finite element methods for pseudo-hyperbolic integro-differential optimal control problems

C. Xu
School of Mathematics and Statistics, Beihua University, Jilin, China
Keywords: псевдогиперболические интегро-дифференциальные уравнения, задачи оптимального управления, априорные оценки ошибки, сверхсходимость, положительно определенные смешанные методы расщепления конечных элементов, pseudo-hyperbolic integro-differential 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 pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas 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 co-state variables. Second, we obtain a superconvergence result for the control variable.

(m, k)-schemes for stiff systems of ODEs and DAEs

A.I. Levykin1,2, A.E. Novikov3, E.A. Novikov3,4
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
3Siberian Federal University, Krasnoyarsk, Russia
4Institute of Computational Mathematicsof the Federal Research Centerof the Krasnoyarsk Scientific Center,Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia
Keywords: методы типа Розенброка, дифференциально-алгебраические уравнения, жесткие системы ОДУ, Rosenbrock-type methods, differential-algebraic equations, stiff systems of ODEs

Abstract >>
This paper deals with the derivation of the optimal form of the Rosenbrock-type methods in terms of the number of non-zero parameters and computational costs per step. A technique of obtaining ( m, k )-methods from the well-known Rosenbrock-type 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.

Convergence of numerical spectral models of the sea surface undulation

K.V. Litvenko1, S.M. Prigarin1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: simulation of random fields, spectral models, convergence, sea surface simulation, extreme ocean waves, rogue waves

Abstract >>
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 spatial-temporal and spatial models.

The finite-difference scheme for one-dimensional maxwell's equations

A.F. Mastryukov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Akad. Lavrentieva 6, Novosibirsk, 630090
Keywords: конечно-разностный, оптимальный, точность, метод Лагерра, электромагнитные, система линейных уравнений, итерации, finite difference, optimal, accuracy, Laguerre method, electromagnetic, linear equations system, iterations

Abstract >>
This paper deals with a difference scheme of second order of approximation for one-dimensional 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 20-25 per cent as compared to the fourth order difference scheme.

Fourth-order numerical scheme based on half-step non-polynomial spline approximations for 1D quasi-linear parabolic equations

R.K. Mohanty1, S. Sharma2
1Department of Applied Mathematics, Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, India
2Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India
Keywords: квазилинейные параболические уравнения, сплайн в напряжении, обобщенное уравнение Бюргерса-Хаксли, обобщенное уравнение Бюргерса-Фишера, итерационный метод Ньютона, quasi-linear parabolic equations, spline in tension, generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, Newton's iterative method

Abstract >>
In this article, we discuss a fourth-order accurate scheme based on non-polynomial spline in tension approximations for the solution of quasi-linear parabolic partial differential equations. The proposed numerical method requires only two half-step points and a central point on a uniform mesh in the spatial direction. This method is derived directly from a continuity condition of the first-order derivative of a non-polynomial 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 Burgers-Huxley equation, the generalized Burgers-Fisher equation, and Burgers' equations in polar coordinates.

Classification of difference schemes of the maximum possible accuracy on extended symmetric stencils for the Schrödinger equation and the heat transfer equation

V.I. Paasonen1,2
1Institute of Computational Technologies SB RAS, Novosibirsk, Russia
2Novosibirsk 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, multi-point scheme, multi-point stencil

Abstract >>
We study all possible symmetric two-level 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.