Investigation of the stability and convergence of difference schemes for the three-dimensional equations of the atmospheric boundary layer
Abstract
In this article we construct a finite-difference scheme for the three-dimensional equations of the atmospheric boundary layer. The solvability of the mathematical model is proved and quality properties of the solutions are studied. A priori estimates are derived for the solution of the differential equations. The mathematical questions of the difference schemes for the equations of the atmospheric boundary layer are studied. Nonlinear terms are approximated such that the integral term of the identity vanishes when it is scalar multiplied. This property of the difference scheme is formulated as a lemma. Main a priori estimates for the solution of the difference problem are derived. Approximation properties are investigated and the theorem of convergence of the difference solution to the solution of the differential problem is provedReferences
A. A. Samarskiy, E.S. Nikolayev, “Metody resheniya setochnykh uravneniy,” Nauka, 1978, pp.420–592.
V. C. Loukopoulos, G. T.Messaris, G. C. Bourantas, “Numerical solution of the incompressible Navier–Stokes equations in primitive variables and velocity–vorticity formulation,” Applied Mathematics and Computation, 222, Oct. 2013, pp. 575-588.
K. Wang, “Iterative schemes for the non-homogeneous Navier–Stokes equations based on the finite element approximation,” Computers & Mathematics with Applications, 71, Jan. 2016, pp. 120-132.
A. Veneziani, Ch. Vergara, “An approximate method for solving incompressible Navier–Stokes problems with flow rate conditions,” Computer Methods in Applied Mechanics and Engineering, 196, Feb. 2007, pp. 1685-1700.
X. Zhang., M. Chen, C.S. Chen, Li. Zhiyong, “Localized method of approximate particular solutions for solving unsteady Navier–Stokes problem,” Applied Mathematical Modelling, 40, Feb. 2016, pp. 2265-2273.
C. Taylor, P. Hood, “A numerical solution of the Navier-Stokes equations using the finite element technique,” Computers & Fluids, 1, Jan. 1973, pp. 73-100.
J. Feifei, L. Jian, Ch. Zhangxin, Z. Zhonghua, “Numerical analysis of a characteristic stabilized finite element method for the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions,” Journal of Computational and Applied Mathematics, 320, Aug. 2017, pp. 43-60.
V. M. Kovenya, “Ob ustoychivosti skhem rasshchepleniya i priblizhennoy faktorizatsii dlya resheniya sistem mnogomernykh uravneniy,” Vychislitel'nyye tekhnologii. 6, 2011, pp. 38-47.
A. Smolarz, V. Lytvynenko, O. Kozhukhovskaya, K. Gromaszek, “Combined clonal negative selection algorithm for diagnostics of combustion in individual PC burner,” Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska, 4, 2013, pp. 69-73.
V. Mashkov, A. Smolarz, V. Lytvynenko, K. Gromaszek, “The problem of system fault-tolerance,” Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska, 4, 2014, pp. 41-44.
G. M. Kobel'kov, “Simmetrichnyye approksimatsii uravneniy Nav'ye–Stoksa,” Matem. sb., 193(7), 2002, pp. 87–108.
A. V. Drutsa, G. M. Kobel'kov, “O skhodimosti raznostnykh skhem dlya uravneniy dinamiki okeana”, Matem. sb., 203(8), 2012, pp. 17–38.
N. I. Sidnyayev, N. M. Gordeyeva, “O tochnosti raznostnoy skhemy dlya uravneniy Nav'ye–Stoksa,” Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(34), 2014, pp. 156–167.
J. Blascoa, R. Codinab, “Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations,” Applied Numerical Mathematics, 38, 2001, pp. 475–497.
H. Guoliang, H. Yinnian, Z. Chen, “A penalty finite volume method for the transient Navier–Stokes equations,” Applied Numerical Mathematics 58, 2008, pp. 1583–1613.
J. de Frutos., B. García., J. Novo, “Two-Grid Mixed Finite-Element Approximations to the Navier-Stokes Equations,” Journal of Scientific Computing, 52, Sept. 2012, pp. 619-637.
A. N. Temirbekov, N. T. Danaev, E. A. Malgazhdarov, “Modeling of Polutants in the Atmosphere Based on Photochemical Reactions,” Eurasian chemico-technological journa., Vol. 16, 2014, pp. 61-71.
O. A. Ladyzhenskaja, “ Matematicheskie voprosy dinamiki vjazkoj neszhimaemoj zhidkosti”, Nauka, Moscow, 1970, pp 27-243
O. M. Belocerkovskij, “Chislennoe modelirovanie v mehanike sploshnyh sred, Fizmatlit,” Moscow, 1994, pp. 34-321.
I. Manak, W. Wójcik, V. Firago, P. Komada “Pomiar stężenia CO z wykorzystaniem metod TDLAS w bliskiej podczerwieni,” Przegląd Elektrotechniczny, 3, 2008, pp. 238-240.
S. Cięszczyk, P. Komada, A. Akhmetova, A. Mussabekova, “Metoda analizy widm mierzonych z wykorzystaniem spektrometrów OP-FTIR w monitorowaniu powietrza atmosferycznego oraz gazów w procesach przemysłowych,” Rocznik Ochrona Środowiska, 18, 2016, pp. 218-234.
D. Sawicki, A. Kotyra, “Porównanie wybranych metod wyznaczania obszaru płomienia w wizyjnym systemie diagnostycznym,” Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska, 4, 2013, pp. 14-17.
Downloads
Published
Issue
Section
License
Copyright (c) 2018 International Journal of Electronics and Telecommunications
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
1. License
The non-commercial use of the article will be governed by the Creative Commons Attribution license as currently displayed on https://creativecommons.org/licenses/by/4.0/.
2. Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. The undersigned also warrants that the manuscript (or its essential substance) has not been published other than as an abstract or doctorate thesis and has not been submitted for consideration elsewhere, for print, electronic or digital publication.
3. User Rights
Under the Creative Commons Attribution license, the author(s) and users are free to share (copy, distribute and transmit the contribution) under the following conditions: 1. they must attribute the contribution in the manner specified by the author or licensor, 2. they may alter, transform, or build upon this work, 3. they may use this contribution for commercial purposes.
4. Rights of Authors
Authors retain the following rights:
- copyright, and other proprietary rights relating to the article, such as patent rights,
- the right to use the substance of the article in own future works, including lectures and books,
- the right to reproduce the article for own purposes, provided the copies are not offered for sale,
- the right to self-archive the article
- the right to supervision over the integrity of the content of the work and its fair use.
5. Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
6. Termination
This agreement can be terminated by the author or the Journal Owner upon two months’ notice where the other party has materially breached this agreement and failed to remedy such breach within a month of being given the terminating party’s notice requesting such breach to be remedied. No breach or violation of this agreement will cause this agreement or any license granted in it to terminate automatically or affect the definition of the Journal Owner. The author and the Journal Owner may agree to terminate this agreement at any time. This agreement or any license granted in it cannot be terminated otherwise than in accordance with this section 6. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.
7. Royalties
This agreement entitles the author to no royalties or other fees. To such extent as legally permissible, the author waives his or her right to collect royalties relative to the article in respect of any use of the article by the Journal Owner or its sublicensee.
8. Miscellaneous
The Journal Owner will publish the article (or have it published) in the Journal if the article’s editorial process is successfully completed and the Journal Owner or its sublicensee has become obligated to have the article published. Where such obligation depends on the payment of a fee, it shall not be deemed to exist until such time as that fee is paid. The Journal Owner may conform the article to a style of punctuation, spelling, capitalization and usage that it deems appropriate. The Journal Owner will be allowed to sublicense the rights that are licensed to it under this agreement. This agreement will be governed by the laws of Poland.
By signing this License, Author(s) warrant(s) that they have the full power to enter into this agreement. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.
