An asymptotic preserving scheme for capturing concentrations in age-structured models arising in adaptive dynamics

Abstract

We propose an asymptotic preserving (A-P) scheme for a population model structured by age and a phenotypical trait with or without mutations. As proved in [26], Dirac concentrations on particular phenotypical traits appear in the case without mutations, which makes the numerical resolution of the problem challenging. Inspired by its asymptotic behaviour, we apply a proper Wentzel-Kramers-Brillouin (WKB) representation of the solution to derive an A-P scheme, with which we can accurately capture the concentrations on a coarse, ε-independent mesh. The scheme is thoroughly analysed and important properties, including the A-P property, are rigorously proved. Furthermore, we observe nearly spectral accuracy in time in our numerical simulations. Next, we generalize the A-P scheme to the case with mutations, where a nonlinear Hamilton-Jacobi equation will be involved in the limiting model as ε→0. It can be formally shown that the generalized scheme is A-P as well, and numerical experiments indicate that we can still accurately solve the problem on a coarse, ε-independent mesh in the phenotype space.