maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2016-3-334-341mais-347Research ArticleОригинальные статьиArticlesАналитико-численный подход для решения сигулярно возмущенных параболиче- ских уравнений с использованием динамически адаптированных сетокAnalytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted MeshesЛукьяненкоД. В.LukyanenkoD. V.

канд. физ.-мат. наук, доцент, 119991, г. Москва, Ленинские горы, МГУ, физический факультет

PhD, associate professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics

lukyanenko@physics.msu.ruВолковВ. Т.VolkovV. T.

канд. физ.-мат. наук, доцент, 119991, г. Москва, Ленинские горы, МГУ, физический факультет

PhD, associate professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics

volkovvt@mail.ruНефедовН. Н.NefedovN. N.

доктор физ.-мат. наук, профессор, 119991, г. Москва, Ленинские горы, МГУ, физический факультет

Professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics

nefedov@phys.msu.ruРекеЛ.ReckeL.

доктор физ.-мат. наук, профессор

Professor

recke@mathematik.hu-berlin.deШнайдерК.SchneiderK.

доктор физ.-мат. наук, профессор

Professor

schneider@wias-berlin.deМосковский государственный университет им. М.В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian FederationHU Berlin, Institut fuЁr Mathematik, Rudower Chaussee, Berlin, GermanyРоссияHU Berlin, Institut fЁur Mathematik, Rudower Chaussee, Berlin, GermanyRussian FederationWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyРоссияWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyRussian Federation201620062016233334341Copyright © Лукьяненко Д.В., Волков В.Т., Нефедов Н.Н., Реке Л., Шнайдер К., 20162016Лукьяненко Д.В., Волков В.Т., Нефедов Н.Н., Реке Л., Шнайдер К.Lukyanenko D.V., Volkov V.T., Nefedov N.N., Recke L., Schneider K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/347

Основной целью данной работы является представление нового аналитико-численного подхода к исследованию сингулярно возмущенных моделей типа реакция-диффузия-адвекция, решения которых содержат движущиеся внутренние переходные слои (фронты). В работе описаны некоторые методы построения динамически адаптированных сеток для эффективного численного решения задач указанного типа. Эти методы основаны на использовании априорной информации о свойствах движущегося фронта, полученной в результате асимптотического анализа. В частности, при построении сетки учитываются априорные асимптотические оценки локализации и скорости фронта, его ширина и структура. Предложенные алгоритмы позволяют существенно снизить затраты вычислительных ресурсов и повысить стабильность численного счета по сравнению с известными классическими подходами. Статья публикуется в авторской редакции.

The main objective of the paper is to present a new analytic-numerical approach to singularly perturbed reaction-diffusion-advection models with solutions containing moving interior layers (fronts). We describe some methods to generate the dynamic adapted meshes for an efficient numerical solution of such problems. It is based on a priori information about the moving front properties provided by the asymptotic analysis. In particular, for the mesh construction we take into account a priori asymptotic evaluation of the location and speed of the moving front, its width and structure. Our algorithms significantly reduce the CPU time and enhance the stability of the numerical process compared with classical approaches.The article is published in the authors’ wording.

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