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.

翻译：本文提出一个用于离散化一类约束Hamilton-Jacobi方程的计算框架，该系统将Hamilton-Jacobi方程与由约束条件确定的拉格朗日乘子耦合。该方程是非局部的，且约束具有有界变差性质。我们证明，在一组一般性假设下，通过有限差分单调格式获得的逼近收敛于约束Hamilton-Jacobi方程的粘性解。约束Hamilton-Jacobi方程常出现在定量遗传学中种群模型的长时与小突变渐近极限问题中。作为实例，我们详细阐述了积分Lotka-Volterra方程极限的格式构造方法。我们还针对渐近区域外的模型构造并分析了一种渐进保持（AP）格式，证明其在向渐近区域过渡过程中保持稳定性。通过数值模拟对格式的理论分析进行展示与讨论，并利用该AP格式推测当环境随时间变化时积分Lotka-Volterra方程的渐近行为。