High-order Discontinuous Galerkin Methods for a class of transport equations with structured populations

In this paper we analyze a discontinuous Galerkin finite element method for approximating solutions to transport equations with certain nonlinearities. We consider models for age-structured populations allowing for a nonlinear removal rate with non-local boundary conditions on the in-flow boundary. The method employs a stabilizing term over the interior edges allowing for convergence in a stronger than usual norm. We establish convergence rates for general higher order basis functions and provide numerical examples consistent with this result.

[1]  Morton E. Gurtin,et al.  On the optimal harvesting of age-structured populations: Some simple models☆ , 1981 .

[2]  Jim Douglas,et al.  Numerical methods for a model of population dynamics , 1987 .

[3]  Graeme Fairweather,et al.  A Box Method for a Nonlinear Equation of Population Dynamics , 1991 .

[4]  Eun-Jae Park,et al.  An upwind scheme for a nonlinear model in age-structured population dynamics , 1995 .

[5]  Mi-Young Kim,et al.  A Collocation Method for the Gurtin-MacCamy Equation with Finite Life-Span , 2001, SIAM J. Numer. Anal..

[6]  Donato Trigiante,et al.  A hybrid scheme for solving a model of population dynamics , 1982 .

[7]  Morton E. Gurtin,et al.  On the optimal harvesting of persistent age-structured populations , 1981 .

[8]  Mi-Young Kim,et al.  AGE-TIME DISCONTINUOUS GALERKIN METHOD FOR THE LOTKA-MCKENDRICK EQUATION , 2003 .

[9]  Mi-Young Kim,et al.  Age-time continuous Galerkin methods for a model of population dynamics , 2009 .

[10]  M. Iannelli,et al.  Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission , 1992 .

[11]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[12]  Mi-Young Kim,et al.  hp-DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA-MCKENDRICK EQUATION: A NUMERICAL STUDY , 2007 .

[13]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[14]  Todd Arbogast,et al.  A finite difference method for a two-sex model of population dynamics , 1989 .

[15]  Chung-Ki Cho,et al.  Second-order accurate difference methods for a one-sex model of population dynamics , 1993 .

[16]  T. Dupont.  Galerkin Methods for First Order Hyperbolics: An Example , 1973 .

[17]  Todd F. Dupont,et al.  Galerkin Methods in Age and Space for a Population Model with Nonlinear Diffusion , 1998, SIAM J. Numer. Anal..

[18]  F A Milner,et al.  Rapidly converging numerical algorithms for models of population dynamics , 1992, Journal of mathematical biology.

[19]  Henry L. Langhaar,et al.  General population theory in the age-time continuum , 1972 .

[20]  Mimmo Iannelli,et al.  On the approximation of the Lotka-McKendrick equation with finite life-span , 2001 .

[21]  J. Reddy.  An introduction to the finite element method , 1989 .

[22]  Mi-Young Kim,et al.  Discontinuous Galerkin Methods for a Model of Population Dynamics with Unbounded Mortality , 2005, SIAM J. Sci. Comput..

[23]  Mimmo Iannelli,et al.  Global boundedness of the solutions to a Gurtin-MacCamy system , 2002 .

[24]  Morton E. Gurtin,et al.  Non-linear age-dependent population dynamics , 1974 .

[25]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[26]  Michael C Mackey,et al.  Dynamic hematological disease: a review , 2009, Journal of mathematical biology.

[27]  Endre Süli,et al.  DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS , 2004 .

[28]  Mi-Young Kim,et al.  DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA¿MCKENDRICK EQUATION WITH FINITE LIFE-SPAN , 2006 .

[29]  L. F. Murphy,et al.  Optimal harvesting of an age-structured population , 1990 .