Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Numerical Analysis and Its Applications (NAA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10187))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    MATH  Google Scholar 

  2. O’Riordan, E., Shishkin, G.: Singularly perturbed parabolic problems with non-smooth data. J. Comput. Appl. Math. 1, 233–245 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Franz, S., Kopteva, N.: Green’s function estimates for a singularly perturbed convection-diffusion problem. J. Differ. Equ. 252, 1521–1545 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hairer, E., Wanner, G.: Solving of Ordinary Differential Equations. Stiff and Differential-Algebraic Problems. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  16. Kalitkin, N., Al’shin, A., Al’shina, E., Rogov, B.: Computations on Quasi-Uniform Grids. Fizmatlit, Moscow (2005). (in Russian)

    Google Scholar 

  17. Rosenbrock, H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5(4), 329–330 (1963)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Faculty of Physics, Department of Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia

    Dmitry Lukyanenko, Nikolay Nefedov, Egor Nikulin & Vladimir Volkov

Authors
  1. Dmitry Lukyanenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolay Nefedov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Egor Nikulin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vladimir Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolay Nefedov .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. Department of Applied Mathematics and Statistics, University of Rousse, Ruse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Lukyanenko, D., Nefedov, N., Nikulin, E., Volkov, V. (2017). Use of Asymptotics for New Dynamic Adapted Mesh Construction for Periodic Solutions with an Interior Layer of Reaction-Diffusion-Advection Equations. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science(), vol 10187. Springer, Cham. https://doi.org/10.1007/978-3-319-57099-0_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57099-0_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57098-3

  • Online ISBN: 978-3-319-57099-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation