Skip to main content
SpringerLink
Log in

Local Discontinuous Galerkin Methods for Reaction-Diffusion Systems on Unstructured Triangular Meshes

  • 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:

  • 1782 Accesses

Abstract

In this paper, on two-dimension unstructured meshes, a fully-discrete scheme is presented for the reaction-diffusion systems, which are often used as mathematical models for many biological, physical and chemical applications. By using local discontinuous Galerkin (LDG) method, the scheme can derive the numerical approximations not only for solutions but also for their gradients at the same time. In addition, the scheme employs the implicit integration factor (IIF) method for temporal discretization, which allows us to take the time-step as \(\delta t=O(h_{min})\), and can be computed element by element, so that it reduces the computational cost greatly. Numerical simulations for the chlorite-iodide-malonic acid (CIMA) model demonstrate the expected behavior of the solutions, the efficiency and advantages of the proposed scheme.

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. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, S.Q., Zhang, Y.T.: Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontnuous Galerkin methods. J. Comput. Phys. 230, 4336–4352 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion system. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dehghan, M., Abbaszadeh, M.: Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction-diffusion system with and without cross-diffusion. Comput. Methods Appl. Mech. Eng. 300, 770–797 (2016)

    Article  MathSciNet  Google Scholar 

  5. Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)

    Article  MATH  Google Scholar 

  6. Lengyel, I., Epstein, I.R.: Modelling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system. Science 251, 650–652 (1991)

    Article  Google Scholar 

  7. Li, B.Q.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, pp. 105–156. Springer, London (2006)

    Google Scholar 

  8. Li, Q., Zheng, C.G., Wang, N.C., Shi, B.C.: LBGK simulations of Turing patterns in CIMA model. J. Sci. Comput. 16, 121–134 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nie, Q., Zhang, Y.T., Zhao, R.: Efficient semi-implicit schemes for stiff systems. J. Comput. Phys. 214, 512–537 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Schnakenberg, J.: Simple chemical reaction systems with limit cycle behavior. J. Theoret. Biol. 81, 389–400 (1979)

    Article  MathSciNet  Google Scholar 

  11. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  12. Zhang, R.P., Yu, X.J., Zhu, J., Loula, A.F.D.: Direct discontinuous Galerkin method for nonlinear reaction-diffusion systems in pattern formulation. Appl. Math. Model. 38, 1612–1621 (2014)

    Article  MathSciNet  Google Scholar 

  13. Zhu, J.F., Zhang, Y.T., Newman, S.A.: Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology. J. Sci. Comput. 40, 391–418 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38, 29–43 (1983)

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the Natural Science Foundation of China (Grant No. 11571002), the Science and Technology Development Foundation of CAEP (Grant Nos. 2013A0202011 and 2015B0101021) and the Defense Industrial Technology Development Program (Grant No. B1520133015).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Na An

  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Na An, Xijun Yu, Chaobao Huang & Maochang Duan

  3. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, China

    Chaobao Huang

Authors
  1. Na An
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xijun Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chaobao Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Maochang Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xijun Yu .

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

An, N., Yu, X., Huang, C., Duan, M. (2017). Local Discontinuous Galerkin Methods for Reaction-Diffusion Systems on Unstructured Triangular Meshes. 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_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57099-0_16

  • 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