  In this paper, we introduce a framework for the discretization of a class of
constrained Hamilton-Jacobi equations, a system coupling a Hamilton-Jacobi
equation with a Lagrange multiplier determined by the constraint. The equation
is non-local, and the constraint has bounded variations. We show that, under a
set of general hypothesis, the approximation obtained with a finite-differences
monotonic scheme, converges towards the viscosity solution of the constrained
Hamilton-Jacobi equation.
  Constrained Hamilton-Jacobi equations often arise as the long time and small
mutation asymptotics of population models in quantitative genetics. As an
example, we detail the construction of a scheme for the limit of an integral
Lotka-Volterra equation. We also construct and analyze an Asymptotic-Preserving
(AP) scheme for the model outside of the asymptotics. We prove that it is
stable along the transition towards the asymptotics.
  The theoretical analysis of the schemes is illustrated and discussed with
numerical simulations. The AP scheme is also used to conjecture the asymptotic
behavior of the integral Lotka-Volterra equation, when the environment varies
in time.
