Numerical analysis of linearly implicit Euler-Riemann method for nonlinear Gurtin-MacCamy model

In this paper, we deal with the existence and stability of an equilibrium age distribution of the linearly implicit Euler-Riemann method for nonlinear age-structured population model with density dependence, i.e., the Gurtin-MacCamy models. It is shown that a dynamical invariance is replicated by numerical solutions for a long time. With the help of infinite-dimensional Leslie operators, the numerical processes are embedded into a nonlinear dynamical process in an infinite dimensional space, which provides a numerical basic reproduction function and numerical endemic equilibrium distributions. As an application to the Logistic model, a numerical reproduction number R₀^h ensures the global stability of disease-free equilibrium whenever R₀^h<1 and the existence of the numerical endemic equilibrium for R₀^h>1. Moreover, instead of the convergence of numerical solutions, it is much more interesting that the numerical solutions preserve the existence of endemic equilibrium for small stepsize, since the numerical reproduction numbers, numerical endemic equilibrium and distribution converge to the exact ones with accuracy of order 1. Finally, some numerical experiments illustrate the verification and the efficiency of our results.