Numerical Analysis of a Pollution and Environment Interaction Model

  • Ivan Dimov
  • Juri KandilarovEmail author
  • Venelin Todorov
  • Lubin Vulkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11189)


Finite difference and finite element approximations for solving numerically the systems of partial differential equations, by which comprehensive models for studying complex environmental problems are studied, are proposed and discussed in this paper. First, we establish a minimum principle for the differential problem and then nonnegativity of the semidiscrete solutions. Algorithms of explicit-implicit and fully explicit schemes are realized for solution of the discrete systems. Numerical experiments are provided to illustrate the efficiency of the algorithms.


Environment interaction model Parabolic system Finite difference scheme FEM Immersed interface method 



The first, third and fourth authors are supported by the Bulgarian National Fund of Science under Projects DN 12/5-2017 and DN 12/4-2017, and the second author - by the Bilateral Project DNTS/Russia 02/12 from 2018.


  1. 1.
    Bratus, A., Mescherin, A., Novozhilov, A.: Mathematical models of interaction between pollutant and environment. In: Proceedings of the International Conference on “Control of Oscillations and Chaos’2000”, vol. 3, St. Petersburg (2000)Google Scholar
  2. 2.
    Bratus, A., Mescherin, A., Novozhilov, A.: Mathematical models of interaction between pollutant and environment, Vest. MGU, Vych. Mat. Kybern. vol. 1, pp. 23–28 (2001)Google Scholar
  3. 3.
    Ganev, K.G., Syrakov, D.E., Zlatev, Z.: Effective indices for emissions from road transport. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2007. LNCS, vol. 4818, pp. 401–409. Springer, Heidelberg (2008). Scholar
  4. 4.
    Hundstorfer, W., Vrewer, J.G.: Numerical Solution of Time-Dependent Advection-Diffuison-Reaction Equations, vol. 33. Springer, Heidelberg (2003). Scholar
  5. 5.
    Kandilarov, J.D., Vulkov, L.G.: The immersed interface method for two-dimansional heat-diffusion equation with singular own sources. Appl. Numer. Math. 57, 486–497 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Li, Z., Ito, K.: The Immersed Interface Method - Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM, Philadelphia (2006)CrossRefGoogle Scholar
  7. 7.
    Marchuk, G.I.: Mathematical Modeling in Environmental Problems. Science, Moscow (1977). (in Russian)Google Scholar
  8. 8.
    Pao, C.V.: Nonlinaer Parabolic and Elliptic Equations. Plenum Press, New York (1992)Google Scholar
  9. 9.
    Penenko, A.V.: Consistent numerical scheme for solving nonlinear inverse source problems with gradient-type algorithms and Newton-Kantorovich methods. Numer. Anal. Appl. 11(1), 73–88 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Thomée, V.: On positivity preservation in some finite element methods for the heat equation. In: Dimov, I., Fidanova, S., Lirkov, I. (eds.) NMA 2014. LNCS, vol. 8962, pp. 13–24. Springer, Cham (2015). Scholar
  11. 11.
    Zlatev, Z., Dimov, I.: Computational and Numerical Challenges in Environmental Modeling. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ivan Dimov
    • 1
  • Juri Kandilarov
    • 2
    Email author
  • Venelin Todorov
    • 1
    • 3
  • Lubin Vulkov
    • 2
  1. 1.Institute of Information and Communication Technologies, BASSofiaBulgaria
  2. 2.Department of MathematicsUniversity of RuseRuseBulgaria
  3. 3.Institute of Mathematics and Informatics, BASSofiaBulgaria

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