vmaitComputational Mathematics and Information TechnologiesComputational Mathematics and Information Technologies2587-8999Донской государственный технический университет10.23947/2587-8999-2023-7-2-7-18vmait-96Research ArticleComputational Mathematics (Вычислительная математика)Разрывный метод Галеркина и его реализация в программном комплексе РАМЕГ3DThe Discontinuous Galerkin Method and its Implementation in the RAMEG3D Software Packagehttps://orcid.org/0000-0001-7295-7002ТишкинВ. Ф.TishkinV. F.

Тишкин Владимир Федорович, член-корреспондент РАН, профессор, доктор физико-математических наук

tishkin@imamod.ru

AuthorID: 110

125047, г. Москва, Миусская пл., 4

 

Vladimir F. Tishkin, Corresponding Member of the Russian Academy of Sciences, Professor, Doctor of Physical and Mathematical Sciences

tishkin@imamod.ru

AuthorID: 110

4, Miusskaya Sq., Moscow, 125047

v.f.tishkin@mail.ruhttps://orcid.org/0000-0001-7596-1672ЛадонкинаМ. Е.LadonkinaM. E.

Ладонкина Марина Евгеньевна, старший научный сотрудник, кандидат физико-математических наук

AuthorID: 134125

125047, г. Москва, Миусская пл., 4

Marina E. Ladonkina, Senior Researcher, Candidate of Physical and Mathematical Sciences

AuthorID: 134125

4, Miusskaya Sq., Moscow, 125047

ladonkina@imamod.ruИнститут прикладной математики им. М. В. Келдыша Российской академии наукРоссияKeldysh Institute of Applied Mathematics of the Russian Academy of SciencesRussian Federation20231007202372718Copyright © Тишкин В.Ф., Ладонкина М.Е., 20232023Тишкин В.Ф., Ладонкина М.Е.Tishkin V.F., Ladonkina M.E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.cmit-journal.ru/jour/article/view/96

В настоящее время метод Галеркина с разрывными базисными функциями (РМГ) или Discontinuous Galerkin Method (DGM) получил широкое распространение для решения сложных разномасштабных задач математической физики, имеющих важное прикладное значение. При его реализации важным является вопрос о выборе дискретной аппроксимации потоков для вязких членов уравнения Навье-Стокса.

Для успешного применения РМГ на трехмерных неструктурированных сетках необходимо сосредоточить внимание на построении лимитирующих функций, на выборе наилучших дискретных аппроксимаций диффузионных потоков и на применении неявных и итерационных методов решения полученных дифференциально-разностных уравнений.

Исследуются численные схемы первого порядка и схемы РМГ второго порядка с численными потоками Годунова, HLLC, Русанова-Лакса-Фридрихса и гибридными потоками. Для методов высокого порядка точности необходимо использовать схемы высокого порядка по времени.

В работе используется схема Рунге-Кутты третьего порядка. При решении уравнения Навье-Стокса разрывным методом Галеркина уравнения записываются в виде системы уравнений первого порядка.

Currently, the Discontinuous Galerkin Method (DGM) is widely used to solve complex multi-scale problems of mathematical physics that have important applied significance. When implementing it, the question of choosing a discrete approximation of flows for viscous terms of the Navier-Stokes equation is important.

It is necessary to focus on the construction of limiting functions, on the selection of the best discrete approximations of diffusion flows, and on the use of implicit and iterative methods for solving the obtained differential-difference equations for the successful application of DGM on three-dimensional unstructured grids.

First-order numerical schemes and second-order DGM schemes with Godunov, HLLC, Rusanov-Lax-Friedrichs numerical flows and hybrid flows are investigated. For high-order precision methods, it is necessary to use high-order time schemes.

The Runge-Kutta scheme of the third order is used in the work. The equations are written as a system of first-order equations, when solving the Navier-Stokes equation by the discontinuous Galerkin method.

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