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Use of Asymptotics for New Dynamic Adapted Mesh Construction for Periodic Solutions with an Interior Layer of Reaction-Diffusion-Advection Equations

  • Dmitry Lukyanenko
  • Nikolay NefedovEmail author
  • Egor Nikulin
  • Vladimir Volkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)

Abstract

This paper presents the development of analytic-numerical approaches to study periodically moving fronts in singularly perturbed reaction-diffusion-advection models. We describe the results of rigorous asymptotic treatment of the problem and suggest a method to generate a dynamic adapted mesh for the numerical solution of such problems. This method based on a priori information. In particular, we take into account a priori estimates on the location of the transition layer, its width and structure. An example is presented to demonstrate the effectiveness of the proposed method.

Keywords

Singularly perturbed parabolic periodic problems Interior layer Shishkin mesh Dynamic adapted mesh 

Notes

Acknowledgements

The work was supported by RFBR (projects No. 17-01-00519, 17-01-00670, 17-01-00159, 16-01-00755 and 16-01-00437) and the Ministry of Education and Science of the Russian Federation.

References

  1. 1.
    Shishkin, G.: Grid approximation of a singularly perturbed quasilinear equation in the presence of a transition layer. Russ. Acad. Sci. Dokl. Math. 47(1), 83–88 (1993)zbMATHGoogle Scholar
  2. 2.
    O’Riordan, E., Shishkin, G.: Singularly perturbed parabolic problems with non-smooth data. J. Comput. Appl. Math. 1, 233–245 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Franz, S., Kopteva, N.: Green’s function estimates for a singularly perturbed convection-diffusion problem. J. Differ. Equ. 252, 1521–1545 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kopteva, N.: Numerical analysis of a 2d singularly perturbed semilinear reaction-diffusion problem. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2008. LNCS, vol. 5434, pp. 80–91. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-00464-3_8 CrossRefGoogle Scholar
  5. 5.
    Kopteva, N., O’Riordan, E.: Shishkin meshes in the numerical solution of singularly perturbed differential equations. Int. J. Numer. Anal. Model. 1(1), 1–18 (2009)zbMATHGoogle Scholar
  6. 6.
    O’Riordan, E., Quinn, J.: Numerical method for a nonlinear singularly perturbed interior layer problem. In: Clavero, C., Gracia, J., Lisbona, F. (eds.) BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods. Lecture Notes in Computational Science and Engineering, vol. 81, pp. 187–195. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    O’Riordan, E., Quinn, J.: Parameter-uniform numerical method for some linear and nonlinear singularly perturbed convection-diffusion boundary turning point problems. BIT Numer. Math. 51(2), 317–337 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kopteva, N., Stynes, M.: Stabilised approximation of interior-layer solutons of a singularly perturbed semilinear reaction-diffusion problem. Numer. Math. 119, 787–810 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nefedov, N., Nikulin, E.: Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem. Russ. J. Math. Phys. 22(2), 215–226 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Volkov, V., Nefedov, N.: Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations. Zh. Vychisl. Mat. Mat. Fiz. 46(4), 615–623 (2006). Comput. Math. Math. Phys. 46(4), 585–593 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Nefedov, N.: Comparison principle for reaction-diffusion-advection problems with boundary and internal layers. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 62–72. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-41515-9_6 CrossRefGoogle Scholar
  12. 12.
    Nefedov, N., Recke, L., Schnieder, K.: Existence and Asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations. J. Math. Anal. Appl. 405, 90–103 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vasileva, A., Butuzov, V., Nefedov, N.: Singularly perturbed problems with boundary and internal layers. Tr. Mat. Inst. Steklova. 268, 268–283 (2010). Proc. Steklov Inst. Math. 268(1), 258–273 (2010)MathSciNetGoogle Scholar
  14. 14.
    Al’shin, A., Al’shina, E., Kalitkin, N., Koryagina, A.: Rosenbrock schemes with complex coefficients for stiff and differential algebraic systems. Comput. Math. Math. Phys. 46(8), 1320–1340 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hairer, E., Wanner, G.: Solving of Ordinary Differential Equations. Stiff and Differential-Algebraic Problems. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  16. 16.
    Kalitkin, N., Al’shin, A., Al’shina, E., Rogov, B.: Computations on Quasi-Uniform Grids. Fizmatlit, Moscow (2005). (in Russian)Google Scholar
  17. 17.
    Rosenbrock, H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5(4), 329–330 (1963)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Dmitry Lukyanenko
    • 1
  • Nikolay Nefedov
    • 1
    Email author
  • Egor Nikulin
    • 1
  • Vladimir Volkov
    • 1
  1. 1.Faculty of Physics, Department of MathematicsLomonosov Moscow State UniversityMoscowRussia

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