Abstract:
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.
This work was supported by RFBR, projects No. 16-01-00755, 14-01-00182, RFBR, project No. 16-01-00437, RFBR
- DFG, project No. 14-01-91333.
Received: 20.05.2016
Bibliographic databases:
Document Type:
Article
UDC:
519.956
Language: English
Citation:
D. V. Lukyanenko, V. T. Volkov, N. N. Nefedov, L. Recke, K. Schneider, “Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of dynamic adapted meshes”, Model. Anal. Inform. Sist., 23:3 (2016), 334–341
\Bibitem{LukVolNef16}
\by D.~V.~Lukyanenko, V.~T.~Volkov, N.~N.~Nefedov, L.~Recke, K.~Schneider
\paper Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of~dynamic~adapted meshes
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 3
\pages 334--341
\mathnet{http://mi.mathnet.ru/mais503}
\crossref{https://doi.org/10.18255/1818-1015-2016-3-334-341}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3520855}
\elib{https://elibrary.ru/item.asp?id=26246299}
Linking options:
https://www.mathnet.ru/eng/mais503
https://www.mathnet.ru/eng/mais/v23/i3/p334
This publication is cited in the following 9 articles:
Qian Yang, Mingkang Ni, “Asymptotics of the Solution to a Stationary Piecewise-Smooth Reaction-Diffusion-Advection Equation”, Chin. Ann. Math. Ser. B, 44:1 (2023), 81
M. A. Davydova, S. A. Zakharova, “Singularly perturbed stationary diffusion model with a cubic nonlinearity”, Differ. Equ., 56:7 (2020), 819–830
N. T. Levashova, N. N. Nefedov, A. V. Yagremtsev, “Existence of a solution in the form of a moving front of a reaction-diffusion-advection problem
in the case of balanced advection”, Izv. Math., 82:5 (2018), 984–1005
A. A. Melnikova, N. N. Derugina, “The dynamics of the autowave front in a model of urban ecosystems”, Mosc. Univ. Phys. Bull., 73:3 (2018), 284–292
D. V. Lukyanenko, M. A. Shishlenin, V. T. Volkov, “Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data”, Commun. Nonlinear Sci. Numer. Simul., 54 (2018), 233–247
E. A. Antipov, N. T. Levashova, N. N. Nefedov, “Asimptoticheskoe priblizhenie resheniya uravneniya reaktsiya-diffuziya-advektsiya s nelineinym advektivnym slagaemym”, Model. i analiz inform. sistem, 25:1 (2018), 18–32
M. A. Davydova, S. A. Zakharova, “Ob odnoi singulyarno vozmuschennoi zadache nelineinoi teploprovodnosti v sluchae sbalansirovannoi nelineinosti”, Model. i analiz inform. sistem, 25:1 (2018), 83–91
M. A. Davydova, N. N. Nefedov, “Existence and Stability of Contrast Structures in Multidimensional Singularly Perturbed Reaction-Diffusion-Advection Problems”, Numerical Analysis and Its Applications, NAA 2016, Lecture Notes in Computer Science, 10187, eds. I. Dimov, I. Farago, L. Vulkov, Springer International Publishing Ag, 2017, 277–285
A. Melnikova, N. Levashova, D. Lukyanenko, “Front Dynamics in An Activator-Inhibitor System of Equations”, Numerical Analysis and Its Applications, NAA 2016, Lecture Notes in Computer Science, 10187, eds. I. Dimov, I. Farago, L. Vulkov, Springer International Publishing Ag, 2017, 492–499
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