On the approximation of the Lotka-McKendrick equation with finite life-span (Q5948580)

The paper is concerned with the numerical approximation of the linear initial-boundary value problem NEWLINE\[NEWLINE\begin{cases} \partial_t u + \partial_a u + \mu(a) u = 0,& a>0,\;t>0,\\ u(0,t) = \int_0^\infty \beta(a) u(a,t) da,& t>0,\\ u(a,0) = u_0(x),& a >0,\end{cases} NEWLINE\]NEWLINE 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, \(a_\dag\), at which the survival probability NEWLINE\[NEWLINE\Pi(a) := \text{ e}^{-\int_0^{a} \mu}NEWLINE\]NEWLINE vanishes. Thanks to the characteristics method, the main problem lies in the numerical approximation of the Cauchy problem for the ordinary differential equation NEWLINE\[NEWLINE\begin{cases} v' = \mu(a) v,& a>0,\\ v(0) = 1.\end{cases} NEWLINE\]NEWLINE 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 \(\Pi(a) = (a_\dag -a)^\lambda\) (\(\lambda>0\)). Effective rates of convergence appear to be better for large \(\lambda s\).