Skip to main content
SpringerLink
Log in

A High Order WENO Scheme for a Hierarchical Size-Structured Population Model

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we develop a high order explicit finite difference weighted essentially non-oscillatory (WENO) scheme for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. The main technical complication is the existence of global terms in the coefficient and boundary condition for this model. We carefully design approximations to these global terms and boundary conditions to ensure high order accuracy. Comparing with the first order monotone and second order total variation bounded schemes for the same model, the high order WENO scheme is more efficient and can produce accurate results with far fewer grid points. Numerical examples including one in computational biology for the evolution of the population of Gambussia affinis, are presented to illustrate the good performance of the high order WENO scheme.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ackleh, A.S., Deng, K., Hu, S.: A quasilinear hierarchical size structured model: well-posedness and approximation. App. Math. Optim. 51, 35–59 (2005)

    Article  MathSciNet  Google Scholar 

  2. Angulo, O., Durán, A., López-Marcos, J.C.: Numerical study of size-structured population models: a case of Gambussia affinis. C. R. Biol. 328, 387–402 (2005)

    Article  Google Scholar 

  3. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blayneh, K.W.: Hierarchical size-structured population model. Dyn. Syst. Appl. 9, 527–540 (2002)

    MathSciNet  Google Scholar 

  5. Calsina, A., Saldana, J.: Asymptotic behavior of a model of hierarchically structured population dynamics. J. Math. Biol. 35, 967–987 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cushing, J.M.: The dynamics of hierarchical age-structured population. J. Math. Biol. 32, 705–729 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cushing, J.M., Li, J.: Juvenile versus adult competition. J. Math. Biol. 29, 457–473 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67, 73–85 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability preserving high order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Henson, S.M., Cushing, J.M.: Hierarchical models of intra-specific competition: scramble versus contest. J. Math. Biol. 34, 755–772 (1996)

    MATH  Google Scholar 

  11. Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kraev, E.A.: Existence and uniqueness for height structured hierarchical population models. Nat. Res. Model. 14, 45–70 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Krumholtz, L.A.: Reproduction in the western mosquitofish, Gambusia affinis (Baird and Girard), and its use in mosquito control. Ecol. Monogr. 18, 1–43 (1948)

    Article  Google Scholar 

  14. Shen, J., Shu, C.-W., Zhang, M.: High resolution schemes for a hierarchical size-structured model. SIAM J. Numer. Anal. 45, 352–370 (2007)

    Article  MathSciNet  Google Scholar 

  15. Shi, J., Hu, C., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002)

    Article  MATH  Google Scholar 

  16. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998)

    Chapter  Google Scholar 

  17. Shu, C.-W.: High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Int. J. Comput. Fluid Dyn. 17, 107–118 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. Weiner, J., Thomas, S.X.: Size variability and competition in plant monocultures. Oikos 47, 211–222 (1986)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China

    Jun Shen & Mengping Zhang

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

Authors
  1. Jun Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chi-Wang Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mengping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chi-Wang Shu.

Additional information

Chi-Wang Shu was supported in part by NSFC grant 10671190 while he was visiting the Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345.

Mengping Zhang was supported in part by NSFC grant 10671190.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, J., Shu, CW. & Zhang, M. A High Order WENO Scheme for a Hierarchical Size-Structured Population Model. J Sci Comput 33, 279–291 (2007). https://doi.org/10.1007/s10915-007-9152-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-007-9152-x

Keywords

Search

Navigation