Abstract

We consider a population structured in phenotypical trait, which influcences the adaptation of individuals to their environment. Each individual has the trait of his parent, up to small mutations. We consider the problem in a regime of long time and small mutations. The distribution of the population is expected to concentrate at some dominant traits. Dominant traits can also evolve in time, thanks to mutations. From a technical point of view, the concentration phenomenon is described thanks to a Hopf-Cole transform in the parabolic model. The asymptotic regime is a constrained Hamilton-Jacobi equation [G. Barles, B. Perthame, 2008 & G. Barles, S. Mirrahimi, B. Perthame, 2009]. The uniqueness of the solution of the limit equation has been adressed very recently [V. Calvez, K.-Y. Lam, 2020]. Because of the lack of regularity of the constraint, it can indeed have jumps, the numerical approximation of the constrained Hamilton-Jacobi equation must be treated with care.  We propose an asymptotic preserving scheme for the problem transformed with Hopf-Cole. We show that it converges outside of the asymptotic regime, and that it enjoys stability properties in the transition to the asymptotic regime. Eventually, we show that the limit scheme is convergent for the constrained Hamilton-Jacobi equation.