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Title: Реализация метода прямых для уравнений в частных производных коллокационными блочными разностными схемами
Authors: Дмитриева, О.А.
Гуськова, Н.Г.
Keywords: уравнения в частных производных
метод прямых
задача Коши
параллельные блочные методы
начальные и граничные условия
Issue Date: Sep-2018
Publisher: Донецький національний технічний університет
Abstract: Данная работа посвящена вопросам получения решений уравнений в частных производных с помощью метода прямых, который представляет собой полудискретный метод c дискретизацией по пространственным переменным, обеспечивающий сведение начальной задачи к задаче Коши, описываемой системой обыкновенных дифференциальных уравнений (СОДУ). Такой подход позволяет достаточно эффективно реализовать большой класс эволюционных уравнений. Рассмотрены вопросы решения полученной СОДУ коллокационными блочными методами, позволяющими обеспечить эффективную параллельную реализацию. При этом все преимущества решения СОДУ (параллельное управление шагом, локальный контроль ошибок, устойчивость решения) реализованы для случая частных производных без значительного увеличения вычислительной сложности. Для тестовых уравнений в частных производных параболического типа с различными типами краевых условий и параметрами жесткости проведены множественные компьютерные эксперименты.
Description: This paper is devoted to the problem of obtaining solutions of partial differential equations using the method of lines, which is a semi-discrete method of discretization in space, which ensures the reduction of the initial problem to the Cauchy problem described by a system of ordinary differential equations (SODU). Such an approach allows us to effectively implement a large class of evolution equations. The problems of solving the received SODU by collocation block methods are considered, allowing to provide an effective parallel implementation. Moreover, all the advantages of the SODU solution (parallel step control, local error control, stability of the solution) are realized for the case of partial derivatives without significant increase in computational complexity. As test equations, one-dimensional parabolic problems were used with different types of boundary conditions and with known exact solutions for estimating the global error of the solution. The spatial discretization was performed by a multi-step multipoint collocation block method with the number of reference and calculated points in the 2 × 2 block. The computer experiments carried out were conditionally divided into several classes. When testing the problems of the first class, the main emphasis was made on the comparative analysis of the accuracy of the solution by the method of lines with respect to known explicit and implicit grid methods with commensurate dimensions of steps in time and space. The procedure for controlling the integration step (τ-refinement) was not used. In the second class of tests, the influence of the variation of the step in space on the magnitude of the global error was considered. It is shown that a shortening of the step with respect to the spatial variable leads to the expected decrease in the global error (by almost an order of magnitude). The third class of experiments was devoted to solving rigid evolutionary problems with the possibility of varying the stiffness parameter. For this class of problems, the advantages of the method of lines are shown. Since all computer experiments were carried out in a sequential mode, it is expected that subsequent experiments in parallel environments using collocation block difference schemes in method of lines, which are initially oriented to a parallel solution, will significantly improve the time exponent. The basic of the material, included in the article, was obtained by the author while working at the SimTech Research Center for Modeling Technologies (SimTech) of the University of Stuttgart. The author is grateful to the director of the SimTech Institute, the president of the International Association of Applied Mathematics and Mechanics, Professor Wolfgang Ehlers for many years of cooperation and support, which contributed to the achievement of new results.
ISSN: 1996-1588
Appears in Collections:Кафедра прикладної математики та інформатики

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