KD-Tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters

In this paper, the question of simulating chemical processes with uncertainty in parameters is considered. A new approach is suggested, which consists in building a dynamic structured net based on a kd-tree, over a space formed by the interval parameters. When the algorithm is executed, during each integration step a piecewise constant polynomial function is build, interpolating the connection between the solution and the exact values of interval parameters. The algorithm has been tested on chemical kinetics problems, including combustion processes, demonstrating its efficiency and wide area of application.

Keywords

interval systems of ODE, dynamic structured grid, Lotka-Volterra model, chemical kinetics

Received

10.11.2018

Publication date

30.11.2018

Cite Download pdf To download PDF you should sign in

GOST	Morozov A., Reviznikov D., Gidaspov V. KD-Tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters // Matematicheskoe modelirovanie – 2018. – V. 30. – Number 12 C. 129-144 [Electronic resource]. URL: http://ras.jes.su/mmod/s207987840001171-7-1-en (circulation date: 19.04.2024). DOI: 10.31857/S023408790001940-8
MLA	Morozov, A., Reviznikov, D., Gidaspov, V. "KD-Tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters." Matematicheskoe modelirovanie 30.12 (2018):129-144. DOI: 10.31857/S023408790001940-8
APA	Morozov A., Reviznikov D., Gidaspov V. (2018). KD-Tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters. Matematicheskoe modelirovanie 30(12), pp.129-144 DOI: 10.31857/S023408790001940-8

Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

Readers community rating: votes 0

1. A.A. Belov, N.N. Kalitkin, L.V. Kuz'mina. Modelirovanie khimicheskoj kinetiki v gazakh // Matematicheskoe modelirovanie, 2016, t.28, №8, s.46–64.

2. R.E. Moore. Interval analysis. Englewood Cliffs: Prentice-Hall, 1966.

3. P. Eijgenraam. The Solution of Initial Value Problems Using Interval Arithmetic: Formulation and Analysis of an Algorithm. Amsterdam : Mathematisch Centrum, 1981, 185 p.

4. R.J. Lohner. Enclosing the solutions of ordinary initial and boundary value problems // Computer Arithmetic: Scientific Computation and Programming Languages, 1987, p.255–286.

5. A.N. Rogalev. Garantirovannye metody resheniya sistem obyknovennykh differentsial'nykh uravnenij na osnove preobrazovaniya simvol'nykh formul // Vychislitel'nye tekhnologii, 2003, t.8, №5, s.102–116;

6. A.N. Rogalev. Garantirovannye otsenki i postroenie mnozhestv dostizhimosti dlya nelinejnykh upravlyaemykh sistem // Sibirskij zhurnal nauki i tekhnologij, 2010, №5, s.148-153;

7. K. Makino, M. Berz. Verified Computations Using Taylor Models and Their Applications // Numerical Software Verification 2017: conference proceedings. (Heidelberg, Germany, July 22-23, 2017). Springer International Publishing AG 2017, p.3-13.

8. M. Berz, K. Makino. Rigorous Reachability Analysis and Domain Decomposition of Taylor Models // Numerical Software Verification 2017: conference proceedings. (Heidelberg, Germany, July 22-23, 2017). Springer International Publishing AG 2017, p.90–97.

9. A.N. Rogalev. Simvol'nye vychisleniya v garantirovannykh metodakh, vypolnennye na neskol'kikh protsessorakh // Vestnik NGU, «Inform. tekhnologii», 2006, t.4, №1, s.56-62;

10. S.P. Sharyj. Interval'nyj analiz ili metody Monte-Karlo? // Vychislitel'nye tekhnologii, 2007, t.12, № 1, s.103-115.

11. B.S. Dobronets. Interval'naya matematika. – Krasnoyarsk: KGU, 2004, 219 s.

12. B.S. Dobronets, E.L. Roschina. Prilozheniya interval'nogo analiza chuvstvitel'nosti // Vychislitel'nye tekhnologii, 2002, t.7, № 1, s.75-82;

13. V.A. Vajtiev, S.A. Mustafina. Poisk oblastej neopredelennosti kineticheskikh parametrov matematicheskikh modelej khimicheskoj kinetiki na osnove interval'nykh vychislenik // Vestnik YuUrGU. Seriya Matematicheskoe modelirovanie i programmirovanie, 2014, t.7, № 2, s.99-110;

14. V.A. Vajtiev, S.A. Mustafina. Poisk oblastej neopredelennosti kineticheskikh parametrov matematicheskikh modelej khimicheskoj kinetiki na osnove interval'nykh vychislenik // Vestnik YuUrGU. Seriya Matematicheskoe modelirovanie i programmirovanie, 2014, t.7, № 2, s.99-110;

15. J. Niesen, T. Hall. On the Global Error of Discretization Methods for Ordinary Differential Equations // Ph.D. Thesis, University of Cambridge, 2004.

16. V.I. Arnol'd. Obyknovennye differentsial'nye uravneniya. Izhevsk: Izhevskaya respublikanskaya tipografiya, 2000, 368 c.

17. V.Yu. Gidaspov, N.S. Severina. Ehlementarnye modeli i vychislitel'nye algoritmy fizicheskoj gazovoj dinamiki / Termodinamika i khimicheskaya kinetika: Uchebnoe posobie. – M.: Faktorial, 2014, 84 s.:

18. Yu. Varnatts, U. Maas, R. Dibbl. Gorenie. Fizicheskie i khimicheskie aspekty, modelirovanie, ehksperimenty, obrazovanie zagryaznyayuschikh veschestv / Per. s angl. G.L. Agafonova. Pod red. P.A. Vlasova. – M.: Fizmatlit, 2003, 352 s.

19. V.P. Glushko, L.V. Gurvich, I.V. Vejts i dr. Termodinamicheskie svojstva individual'nykh veschestv, Tom 1. M.: Nauka, 1978-2004 g.

20. E.A. Novikov, M.I. Golushko. (m, 3)-metod tret'ego poryadka dlya zhestkikh neavtonomnykh sistem ODU// Vychislitel'nye tekhnologii, 1998, t.3, № 3, s.48-54: