maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2017-3-322-338mais-521Research ArticleОригинальные статьиArticlesПостроение динамически адаптированной сетки для эффективного численного решения сингулярно возмущенного уравнения типа реакция-адвекция-диффузияDynamically Adapted Mesh Construction for the Efficient Numerical Solution of a Singular Perturbed Reaction-diffusion-advection EquationЛукьяненкоДмитрий ВитальевичLukyanenkoDmitry V.

кандидат физико-математических наук, физический факультет.

Ленинские горы, д. 1, стр.  2, Москва, ГСП-1, 119991

PhD, Faculty of Physics.

1, bld.  2 Leninskiye Gory, Moscow,  GSP-1, 119991

lukyanenko@physics.msu.ruhttps://orcid.org/0000-0002-4205-4141ВолковВладимир ТарасовичVolkovVladimir T.

кандидат физико-математических наук, физический факультет.

Ленинские горы, д. 1, стр.  2, Москва, ГСП-1, 119991

PhD, Faculty of Physics.

1, bld.  2 Leninskiye Gory, Moscow,  GSP-1, 119991

volkovvt@mail.ruhttps://orcid.org/0000-0002-3651-6434НефедовНиколай НиколаевичNefedovNikolay N.

доктор физико-математических наук, физический факультет.

professor, Dr.  Sci.,Faculty of Physics.

1, bld.  2 Leninskiye Gory, Moscow,  119991

nefedov@phys.msu.ruМосковский государственный университет имени М.В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian Federation201730062017243322338Copyright © Лукьяненко Д.В., Волков В.Т., Нефедов Н.Н., 20172017Лукьяненко Д.В., Волков В.Т., Нефедов Н.Н.Lukyanenko D.V., Volkov V.T., Nefedov N.N.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/521

В данной работе на примере численного решения сингулярно возмущенного уравнения  Бюргерса  мы  рассматриваем метод  построения  динамически  адаптированной сетки, который  позволяет  существенно  улучшить  численный  счет  для  уравнений  такого  типа.  Для  построения  данной сетки мы используем  априорную  информацию, основанную на асимптотическом анализе  исходной задачи.  В частности,  мы используем  информацию о скорости внутреннего  слоя, его толщине и структуре. Предложенный в работе алгоритм  способен существенным  образом упростить численную сложность  решаемой задачи  и улучшить  ее устойчивость  по сравнению с классическими  подходами,  используемыми для  решения  задач  такого  класса.  Приведенный  численный эксперимент  демонстрирует эффективность предложенного метода.

Статья публикуется в авторской  редакции.

This  work develops  a theory  of the  asymptotic-numerical investigation of the  moving fronts  in reaction-diffusion-advection models.  By considering  the  numerical  solution  of the  singularly perturbed Burgers’s  equation  we discuss a method  of dynamically  adapted mesh  construction that is able to significantly  improve  the  numerical  solution  of this  type of equations.  For  the  construction we use a priori information that is based  on the  asymptotic analysis  of the  problem.  In  particular, we take  into account the information about  the speed of the transition layer, its width  and structure. Our algorithms  are able to reduce significantly complexity and enhance stability of the numerical  calculations in comparison  with classical approaches for solving this class of problems.  The numerical  experiment is presented to demonstrate the effectiveness of the proposed  method.

The article  is published  in the authors’  wording.

 

сингулярное возмущениевнутренний слойдинамически адаптированная сеткаsingularly perturbedinterior layerdynamically adapted meshРФФИ (проекты № 16-01-00755, 16-01-00437, 17-51-53002 и 17-01-00159); РФФИ, проект № 16-01-00437RFBR, projects No. 16-01-00755, 16-01-00437, 17-51-5300, 17-01-00159; RFBR, projects No. 16-01-00437ReferencesV. T. Volkov, N. N. Nefedov, “Asymptotic-numerical investigation of generation and motion of fronts in phase transition models”, Lecture Notes in Computer Science, 8236 (2013), 524–531.V. T. Volkov, N. N. Nefedov, “Asymptotic-numerical investigation of generation and motion of fronts in phase transition models”, Lecture Notes in Computer Science, 8236 (2013), 524–531.V. T. Volkov, N. E. Grachev, N. N. Nefedov, A. N. Nikolaev, “On the formation of sharp transition layers in two-dimensional reaction-diffusion models”, Comput. Math. and Math. Phys., 47:8 (2007), 1301–1309.V. T. Volkov, N. E. Grachev, N. N. Nefedov, A. N. Nikolaev, “On the formation of sharp transition layers in two-dimensional reaction-diffusion models”, Comput. Math. and Math. Phys., 47:8 (2007), 1301–1309.E. A. Antipov, N. T. Levashova, N. N. Nefedov, “Asymptotics of the front motion in the reaction-diffusion-advection problem,”, Comput. Math. and Math. Phys., 54:10 (2014), 1536–1549.E. A. Antipov, N. T. Levashova, N. N. Nefedov, “Asymptotics of the front motion in the reaction-diffusion-advection problem,”, Comput. Math. and Math. Phys., 54:10 (2014), 1536–1549.A. B. Al’shin, E. A. Al’shina, N. N. Kalitkin, A. B. Koryagina, “Rosenbrock schemes with complex coefficients for stiff, differential algebraic systems”, Comput. Math. and Math. Phys., 46:8 (2006), 1320–1340.A. B. Al’shin, E. A. Al’shina, N. N. Kalitkin, A. B. Koryagina, “Rosenbrock schemes with complex coefficients for stiff, differential algebraic systems”, Comput. Math. and Math. Phys., 46:8 (2006), 1320–1340.G.I. Shishkin, “Necessary conditions for ε-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers”, Comput. Math. and Math. Phys., 47:10 (2007), 1636–1655.G.I. Shishkin, “Necessary conditions for ε-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers”, Comput. Math. and Math. Phys., 47:10 (2007), 1636–1655.G. I. Shishkin, L. P. Shishkina, P. W. Hemker, “A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique”, Comput. Meth. Appl. Math., 4:1 (2004), 105–127.G. I. Shishkin, L. P. Shishkina, P. W. Hemker, “A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique”, Comput. Meth. Appl. Math., 4:1 (2004), 105–127.G.I. Shishkin, “Grid Approximation of a Singularly Perturbed Parabolic Equation on a Composed Domain with a Moving Interface Containing a Concentrated Source”, Comput. Math. and Math. Phys., 43 (2003), 1738–1755.G.I. Shishkin, “Grid Approximation of a Singularly Perturbed Parabolic Equation on a Composed Domain with a Moving Interface Containing a Concentrated Source”, Comput. Math. and Math. Phys., 43 (2003), 1738–1755.G.I. Shishkin, “Grid approximation of a singularly perturbed quasilinear equation in the presence of a transition layer”, Russian Acad. Sci. Dokl. Math., 47:1 (1993), 83–88.G.I. Shishkin, “Grid approximation of a singularly perturbed quasilinear equation in the presence of a transition layer”, Russian Acad. Sci. Dokl. Math., 47:1 (1993), 83–88.J. Quinn, “A numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location”, Journal of Computational, Applied Mathematics, 290 (2015), 500–515.J. Quinn, “A numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location”, Journal of Computational, Applied Mathematics, 290 (2015), 500–515.E. O’Riordan, J. Quinn, “Numerical method for a nonlinear singularly perturbed interior layer problem”, Lecture Notes in Computational Science and Engineering, 81 (2011), 187–195.E. O’Riordan, J. Quinn, “Numerical method for a nonlinear singularly perturbed interior layer problem”, Lecture Notes in Computational Science and Engineering, 81 (2011), 187–195.E. O’Riordan, J. Quinn, “Parameter-uniform numerical method for some linear, nonlinear singularly perturbed convection-diffusion boundary turning point problems”, BIT Numerical Mathematics, 51:2 (2011), 317–337.E. O’Riordan, J. Quinn, “Parameter-uniform numerical method for some linear, nonlinear singularly perturbed convection-diffusion boundary turning point problems”, BIT Numerical Mathematics, 51:2 (2011), 317–337.N. Kopteva, M. Stynes, “Stabilised approximation of interior-layer solutons of a singularly perturbed semilinear reaction-diffusion problem”, Numerische Mathematik, 119:4 (2011), 787–810.N. Kopteva, M. Stynes, “Stabilised approximation of interior-layer solutons of a singularly perturbed semilinear reaction-diffusion problem”, Numerische Mathematik, 119:4 (2011), 787–810.N. Kopteva, “Numerical analysys of a 2d singularly perturbed semilinear reaction-diffusion problem”, Lecture Notes in Computer Science, 5434 (2009), 80–91.N. Kopteva, “Numerical analysys of a 2d singularly perturbed semilinear reaction-diffusion problem”, Lecture Notes in Computer Science, 5434 (2009), 80–91.S. Franz, N. Kopteva, “Green’s function estimates for a singularly perturbed convectiondiffusion problem”, J. Differential Equations, 252 (2012), 1521–1545.S. Franz, N. Kopteva, “Green’s function estimates for a singularly perturbed convectiondiffusion problem”, J. Differential Equations, 252 (2012), 1521–1545.E. O’Riordan, G.I. Shishkin, “Singularly perturbed parabolic problems with non-smooth data”, J. of Computational, Applied Mathematics, 1 (2004), 233–245.E. O’Riordan, G.I. Shishkin, “Singularly perturbed parabolic problems with non-smooth data”, J. of Computational, Applied Mathematics, 1 (2004), 233–245.P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust computational techniques for boundary layers, Chapman, Hall/CRC, 2000.P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust computational techniques for boundary layers, Chapman, Hall/CRC, 2000.P.A. Farrell, E. O’Riordan, G.I. Shishkin, “A class of singularly perturbed semilinear differential equations with interior layers”, Mathematics of Computation, 74:252 (2005), 1759–1776.P.A. Farrell, E. O’Riordan, G.I. Shishkin, “A class of singularly perturbed semilinear differential equations with interior layers”, Mathematics of Computation, 74:252 (2005), 1759–1776.Natalia Kopteva, Eugene O’Riordan, “Shishkin meshes in the numerical solution of singularly perturbed differential equations”, International Journal of Numerical Analysis, Modeling, 1:1 (2009), 1–18.Natalia Kopteva, Eugene O’Riordan, “Shishkin meshes in the numerical solution of singularly perturbed differential equations”, International Journal of Numerical Analysis, Modeling, 1:1 (2009), 1–18.V. T. Volkov, N. N. Nefedov, E. A. Antipov, “Asymptotic-Numerical Method for Moving Fronts in Two-Dimensional R-D-A Problems”, Lecture Notes in Computer Science, 9045 (2015), 408–416.V. T. Volkov, N. N. Nefedov, E. A. Antipov, “Asymptotic-Numerical Method for Moving Fronts in Two-Dimensional R-D-A Problems”, Lecture Notes in Computer Science, 9045 (2015), 408–416.N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, B. V. Rogov, Computations on QuasiUniform Grids, Fizmatlit, Moscow, 2005, (in Russian).N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, B. V. Rogov, Computations on QuasiUniform Grids, Fizmatlit, Moscow, 2005, (in Russian).

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