On the approximation of the Lotka-McKendrick equation with finite life-span. (English) Zbl 0998.65103

Iannelli, M.; Milner, F. A.

J. Comput. Appl. Math. 136, No. 1-2, 245-254 (2001).

The paper is concerned with the numerical approximation of the linear initial-boundary value problem \[ \begin{cases} \partial_t u + \partial_a u + \mu(a) u = 0,& a>0,\;t>0,\\ u(0,t) = \int_0^\infty \beta(a) u(a,t) da,& t>0,\\ u(a,0) = u_0(x),& a >0,\end{cases} \] where \(\beta(a)\geq 0\) is the birth rate and \(\mu(a)\geq 0\) is the mortality rate. Attention is paid to (realistic) situations when there exists a maximum age, \(a_†\), at which the survival probability \[ \Pi(a) := \text{ e}^{-\int_0^{a} \mu} \] vanishes. Thanks to the characteristics method, the main problem lies in the numerical approximation of the Cauchy problem for the ordinary differential equation \[ \begin{cases} v' = \mu(a) v,& a>0,\\ v(0) = 1.\end{cases} \] Under suitable assumptions on the mortality rate ensuring polynomial bounds on \(\Pi\) and its derivatives, both the Euler (explicit and implicit) and the Crank-Nicolson discretizations are discussed. Error bounds are derived and compared to numerical simulations in the case \(\Pi(a) = (a_† -a)^\lambda\) (\(\lambda>0\)). Effective rates of convergence appear to be better for large \(\lambda s\).

Reviewer: S.Benzoni (Lyon)

MSC:

65M25	Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
92D25	Population dynamics (general)
35L45	Initial value problems for first-order hyperbolic systems

Keywords:

finite life-span populations; Euler scheme; Crank-Nicolson scheme; Lotka-McKendrick equation; linear initial-boundary value problem; characteristic method; error bounds; convergence

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