Some Features of the Asymptotic-Numerical Method for the Moving Fronts Description in Two-Dimensional Reaction-Diffusion Problems

  • Vladimir VolkovEmail author
  • Dmitry Lukyanenko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


This paper develops an analytic-numerical approach for the description of moving fronts in two-dimensional nonlinear singularly perturbed parabolic equations. Asymptotic technique allows to reduce two-dimensional nonlinear reaction-diffusion equation to a series of more simple one-dimensional problems. This decomposition significantly decreases the complexity of numerical calculations and allows the effective use of parallel computing. Some numerical experiments are presented to demonstrate the main features of the proposed method.



This work is supported by RSCF, project No. 18-11-00042.


  1. 1.
    Volkov, V., Nefedov, N., Antipov, E.: Asymptotic-numerical method for moving fronts in two-dimensional R-D-A problems. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) FDM 2014. LNCS, vol. 9045, pp. 408–416. Springer, Cham (2015). Scholar
  2. 2.
    Antipov, E.A., Volkov, V.T., Levashova, N.T., Nefedov, N.N.: Moving front solution of the reaction-diffusion problem. Model. Anal. Inf. Syst. 24(3), 259–279 (2017)CrossRefGoogle Scholar
  3. 3.
    Volkov, V., Nefedov, N.: Asymptotic-numerical investigation of generation and motion of fronts in phase transition models. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 524–531. Springer, Heidelberg (2013). Scholar
  4. 4.
    Volkov, V.T., Grachev, N.E., Nefedov, N.N., Nikolaev, A.N.: On the formation of sharp transition layers in two-dimensional reaction-diffusion models. J. Comp. Math. Math. Phys. 47(8), 1301–1309 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fife, P.C., Hsiao, L.: The generation and propagation of internal layers. Nonlinear Anal. Theory Methods Appl. 12(1), 19–41 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alshin, A.B., Alshina, E.A., Kalitkin, N.N., Koryagina, A.B.: Rosenbrock schemes with complex coefficients for stiff and differential algebraic systems. Comp. Math. Math. Phys. 46(8), 1320–1340 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5(4), 329–330 (1963)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kalitkin, N.N., Alshin, A.B., Alshina, E.A., Rogov, B.V.: Computations on Quasi-Uniform Grids, Fizmatlit, Moscow (2005). (in Russian)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of PhysicsLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations