A convergent numerical scheme to a parabolic equation with a nonlocal boundary condition

TL;DR

This paper introduces a convergent implicit numerical scheme for a nonlinear age-structured PDE with nonlocal boundary conditions, combining characteristics, finite differences, and quadrature methods, validated through theoretical analysis and simulations.

Contribution

It presents a novel implicit scheme for a nonlinear PDE with nonlocal boundary conditions, ensuring convergence and consistency, and demonstrates its effectiveness via numerical simulations.

Findings

The scheme is consistent and convergent.

Numerical simulations validate theoretical results.

The method effectively handles nonlocal boundary conditions.

Abstract

In this paper, a numerical scheme for a nonlinear McKendrick-von Foerster equation with diffusion in age (MV-D) with the Dirichlet boundary condition is proposed. The main idea to derive the scheme is to use the discretization based on the method of characteristics to the convection part, and the finite difference method to the rest of the terms. The nonlocal terms are dealt with the quadrature methods. As a result, an implicit scheme is obtained for the boundary value problem under consideration. The consistency and the convergence of the proposed numerical scheme is established. Moreover, numerical simulations are presented to validate the theoretical results.