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

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.

[1]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[2]  Azmy S. Ackleh,et al.  A Quasilinear Hierarchical Size-Structured Model: Well-Posedness and Approximation , 2005 .

[3]  Jacob Weiner,et al.  Size variability and competition in plant monocultures , 1986 .

[4]  Centro internazionale matematico estivo. Session,et al.  Advanced Numerical Approximation of Nonlinear Hyperbolic Equations , 1998 .

[5]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[6]  Jim M Cushing,et al.  Hierarchical models of intra-specific competition: scramble versus contest , 1996 .

[7]  Jie Shen,et al.  Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains , 2007, SIAM J. Numer. Anal..

[8]  Chi-Wang Shu,et al.  High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD , 2001 .

[9]  Jim M Cushing,et al.  Juvenile versus adult competition , 1991 .

[10]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[11]  Chi-Wang Shu,et al.  A technique of treating negative weights in WENO schemes , 2000 .

[12]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[13]  Kbenesh W. Blayneh,et al.  A hierarchical size-structured population model. , 1996 .

[14]  Oscar Angulo,et al.  Numerical study of size-structured population models: a case of Gambussia affinis. , 2005, Comptes rendus biologies.

[15]  Joan Saldaña,et al.  Asymptotic behaviour of a model of hierarchically structured population dynamics , 1997 .

[16]  E. Kraev,et al.  EXISTENCE AND UNIQUENESS FOR HEIGHT STRUCTURED HIERARCHICAL POPULATION MODELS , 2001 .

[17]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[18]  Louis A. Krumholz,et al.  Reproduction in the Western Mosquitofish, Gambusia affinis affinis (Baird & Girard), and Its Use in Mosquito Control , 1948 .

[19]  Jun Shen,et al.  High Resolution Schemes for a Hierarchical Size-Structured Model , 2007, SIAM J. Numer. Anal..

[20]  Jim M Cushing,et al.  The dynamics of hierarchical age-structured populations , 1994 .

[21]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..