2024/03/19 by Benoît Gaudeul, Gaudeul, Benoît, Hélène Hivert +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2403.12557
openalex created_date 2024/03/19 · openalex publication_date 2024/03/19 · openalex updated_date 2026/08/01
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.