vmaitComputational Mathematics and Information TechnologiesComputational Mathematics and Information Technologies2587-8999Донской государственный технический университет10.23947/2587-8999-2017-2-148-155vmait-88Research ArticleСтатьиЭффективный параллельный программный комплекс для решения уравнений Навье- Стокса разрывным методом ГалеркинаEfficient parallel software system for solving Navier-Stokes equations by the discontinuous Galerkin methodhttps://orcid.org/0000-0001-7988-6323КрасновМихаил МихайловичKrasnovMikhail Mikhailovich

Краснов Михаил Михайлович, старший научный сотрудник, Институт прикладной математики им. М.В. Келдыша Российской академии наук (125047, Москва, Миусская пл., д.4)

Krasnov Mikhail Mikhailovich, Senior Researcher, Institute of Applied Mathematics, Keldysh Institute of Applied Mathematics Russian Academy of Sciences (4 Miusskaya pl., Moscow, Russian Federation)

kmm@kiam.ruhttps://orcid.org/0000-0003-3240-2963КучуговПавел АлександровичKuchugovPavel Aleksandrovich

Kuchugov Pavel Aleksandrovich, Candidate of Science in Physics and Maths, Researcher, KeldyshInstitute of Applied Mathematics Russian Academy of Sciences (4 Miusskaya pl., Moscow, RussianFederation)

pkuchugov@gmail.comhttps://orcid.org/0000-0001-7596-1672ЛадонкинаМарина ЕвгеньевнаLadonkinaMarina Evgenyevna

Ладонкина Марина Евгеньевна, кандидат физико-математических наук, Институт прикладной математики им. М.В. Келдыша Российской академии наук (125047, Москва, Миусская пл., д. 4)

Ladonkina Marina Evgenyevna, Candidate of Science in Physics and Maths, Keldysh Institute ofApplied Mathematics Russian Academy of Sciences (4 Miusskaya pl., Moscow, Russian Federation)

ladonkina@imamod.ruТишкинВладимир ФедоровичTishkinVladimir Fedorovich

Тишкин Владимир Федорович, член-корреспондент РАН, профессор, доктор физико-математических наук, Институт прикладной математики им. М.В. Келдыша Российскойакадемии наук (125047, Москва, Миусская пл., д. 4)

Tishkin Vladimir Fedorovich, Corresponding Member of the Russian Academy of Sciences, Professor, Doctor of Science in Physics and Maths, Institute of Applied Mathematics, Lavrentyev Institute of Hydrodynamics (Siberian Branch of the Russian Academy of Science), (Academician Lavrentiev Avenue, 15, Novosibirsk, Russian Federation)

tishkin@mail.ruИнститут прикладной математики им. М.В. Келдыша Российской академии наук (125047, Москва, Миусская пл., д.4)РоссияKeldysh Institute of Applied Mathematics Russian Academy of Sciences (4 Miusskaya pl., Moscow, Russian Federation)Russian FederationИнститут прикладной математики им. М.В. Келдыша Российской академии наук (125047, Москва, Миусская пл., д.4)РоссияKeldysh Institute of Applied Mathematics Russian Academy of Sciences (4 Miusskaya pl., Moscow, Russian Federation)Russian FederationИнститут прикладной математики им. М.В. Келдыша Российской академии наук (125047, Москва, Миусская пл., д.4)РоссияInstitute of Applied Mathematics, Lavrentyev Institute of Hydrodynamics (Siberian Branch of the Russian Academy of Science), (Academician Lavrentiev Avenue, 15, Novosibirsk, Russian Federation)Russian Federation2017290320231288Copyright © Краснов М.М., Кучугов П.А., Ладонкина М.Е., Тишкин В.Ф., 20232023Краснов М.М., Кучугов П.А., Ладонкина М.Е., Тишкин В.Ф.Krasnov M.M., Kuchugov P.A., Ladonkina M.E., Tishkin V.F.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/88

Реализованы алгоритмы решения уравнения Навье-Стокса на трехмерной тетраэдральной сетке методом Галеркина с разрывными базисными функциями. При создании программного кода использован новый подход к программированию задач математической физики, позволяющий компактно записывать и эффективно реализовывать математические формулы, в частности, за счет введения понятия «сеточного оператора», аналогичного математическому, единообразно реализовывать подход на разных типах сеток и для различных вычислительных архитектур. Исследуется эффективность созданного программного кода. 

Algorithms for solving the Navier-Stokes equations on a three-dimensional tetrahedral grid by the discontinuous Galerkin method were realized. Under the code development a new approach to programming problems of mathematical physics was used which allows one compactly write and effectively implement mathematical expressions in particular due to introduction of the concept of «grid operator» similar to mathematical one and uniformly realize algorithms for various grid types and computing architectures. The efficiency of this numerical code is investigated.

уравнения Навье-Стоксаразрывный метод Галеркинапараллельное программированиешаблонное метапрограммированиеNavier-Stokes equationsdiscontinuous Galerkin methodparallel programmingtemplated meta-programmingРабота выполнена при финансовой поддержке Российского научного фонда (код проекта 16-11-10033).The research is done with the financial support from RSF Projects No. 16-11-10033.ReferencesCockburn B. Introduction to the Discontinuous Galerkin Method for Convection Dominated Problems, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, 1998, v.1697, pp.151-268.Cockburn B. Introduction to the Discontinuous Galerkin Method for Convection Dominated Problems, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, 1998, v.1697, pp.151-268.A.K. Pany and S.Yadav. An hp-Local Discontinuous Galerkin method for Parabolic Integro-Differential Equations, OCCAM, Report No. 09/30.A.K. Pany and S.Yadav. An hp-Local Discontinuous Galerkin method for Parabolic Integro-Differential Equations, OCCAM, Report No. 09/30.Ladonkina M.E., Neklyudova O.A., Tishkin V.F., Utyralov D.I. Realization of the boundary conditions of adhesion for the Galerkin discontinuous method // Prep. M.V. Keldysh, 2014, 16 p.Ladonkina M.E., Neklyudova O.A., Tishkin V.F., Utyralov D.I. Realization of the boundary conditions of adhesion for the Galerkin discontinuous method // Prep. M.V. Keldysh, 2014, 16 p.Ladonkina M.E., Tishkin V.F. On methods of Godunov type of high accuracy order // Reports of the Academy of Sciences, 2015, T.461, No. 4, pp. 390-393.Ladonkina M.E., Tishkin V.F. On methods of Godunov type of high accuracy order // Reports of the Academy of Sciences, 2015, T.461, No. 4, pp. 390-393.Ladonkina M.E., Tishkin V.F. A generalization of the Godunov method, using piecewise polynomial approximations, Differentsial'nye Uravneniya, 2015, Vol. 51, No. 7, pp. 899-907.Ladonkina M.E., Tishkin V.F. A generalization of the Godunov method, using piecewise polynomial approximations, Differentsial'nye Uravneniya, 2015, Vol. 51, No. 7, pp. 899-907.Krasnov M.M. Operator library for solving three-dimensional grid problems of mathematical physics using graphics cards with CUDA architecture. // Matematicheskoe Modelirovanie, 2015, vol.27, no. 3, pp.109-120.Krasnov M.M. Operator library for solving three-dimensional grid problems of mathematical physics using graphics cards with CUDA architecture. // Matematicheskoe Modelirovanie, 2015, vol.27, no. 3, pp.109-120.Bassi F., Rebay S. A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations // Journal of Computational Physics, 1997, 131, pp. 267-279.Bassi F., Rebay S. A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations // Journal of Computational Physics, 1997, 131, pp. 267-279.SK Godunov. Difference method for numerical calculation of discontinuous solutions of hydrodynamics equations // Matem. sb., 1959. 47 (89): 3, pp. 271-306.SK Godunov. Difference method for numerical calculation of discontinuous solutions of hydrodynamics equations // Matem. sb., 1959. 47 (89): 3, pp. 271-306.Rusanov V.V. Calculation of the interaction of non-stationary shock waves with obstacles. // Journal of Computational Mathematics and Mathematical Physics, 1961. Т.I., №2, pp. 267-279.Rusanov V.V. Calculation of the interaction of non-stationary shock waves with obstacles. // Journal of Computational Mathematics and Mathematical Physics, 1961. Т.I., №2, pp. 267-279.Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation // Сommunications on Pure and Applied Mathematics. 1954,7, No. 1, pp. 159 -193.Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation // Сommunications on Pure and Applied Mathematics. 1954,7, No. 1, pp. 159 -193.Arnold D.N., Brezzi F., Cockburn B., Marini L.D. Uni fi ed analysis of discontinuous Galerkin methods for elliptic problems. / / SIAM Journal on Numerical Analysis, 2002, 29, pp. 1749-1779.Arnold D.N., Brezzi F., Cockburn B., Marini L.D. Uni fi ed analysis of discontinuous Galerkin methods for elliptic problems. / / SIAM Journal on Numerical Analysis, 2002, 29, pp. 1749-1779.A.K. Pany and S. Yadav An hp-Local Discontinuous Galerkin method for Parabolic Integro-Differential Equations, OCCAM, Report No. 09/30A.K. Pany and S. Yadav An hp-Local Discontinuous Galerkin method for Parabolic Integro-Differential Equations, OCCAM, Report No. 09/30M.E. Ladonkina, OA Neklyudova, V.F. Tishkin. Investigation of the influence of the limiter on the order of the accuracy of the solution by the Galerkin discontinuous method. // KIAM Preprint. M.V. Keldysh, 2012, No. 34, pp. 31.M.E. Ladonkina, OA Neklyudova, V.F. Tishkin. Investigation of the influence of the limiter on the order of the accuracy of the solution by the Galerkin discontinuous method. // KIAM Preprint. M.V. Keldysh, 2012, No. 34, pp. 31.M.E. Ladonkina, O.A. Neklyudova, V.F. Tishkin. Investigation of the influence of the limiter on the order of accuracy of the solution by the Galerkin discontinuous method, Mat. Model., 2012, Т.24, №12, pp. 124-128.M.E. Ladonkina, O.A. Neklyudova, V.F. Tishkin. Investigation of the influence of the limiter on the order of accuracy of the solution by the Galerkin discontinuous method, Mat. Model., 2012, Т.24, №12, pp. 124-128.M.E. Ladonkina, OA Neklyudova, V.F. Tishkin. High accuracy limiter for the Galerkin discontinuous method on triangular meshes. // KIAM Preprint. M.V. Keldysh, 2013, No. 53, 26c.M.E. Ladonkina, OA Neklyudova, V.F. Tishkin. High accuracy limiter for the Galerkin discontinuous method on triangular meshes. // KIAM Preprint. M.V. Keldysh, 2013, No. 53, 26c.Krasnov MM, Kuchugov PA, Ladonkina ME, Tishkin VF Galerkin discontinuous method on three-dimensional tetrahedral grids. Using the Operator Programming Method. // Mathematical Modeling, 2017, Vol. 29, No. 2, P. 3-22.Krasnov MM, Kuchugov PA, Ladonkina ME, Tishkin VF Galerkin discontinuous method on three-dimensional tetrahedral grids. Using the Operator Programming Method. // Mathematical Modeling, 2017, Vol. 29, No. 2, P. 3-22.

The authors declare that there are no conflicts of interest present.