A numerical scheme for a diffusion equation with nonlocal nonlinear boundary condition

TL;DR

This paper introduces a numerical scheme for solving a diffusion equation with complex nonlocal nonlinear boundary conditions, addressing stability and convergence issues through an abstract discretization theory.

Contribution

The paper develops a novel numerical scheme for the McKendrick-Von Foerster equation with diffusion, incorporating a nonlocal nonlinear boundary condition, and provides a rigorous stability and convergence analysis.

Findings

The scheme is stable under specified conditions.

Convergence of the scheme is proven.

The method effectively handles nonlocal nonlinear boundary conditions.

Abstract

In this article, a numerical scheme to find approximate solutions to the McKendrick-Von Foerster equation with diffusion (M-V-D) is presented. The main difficulty in employing the standard analysis to study the properties of this scheme is due to presence of nonlinear and nonlocal term in the Robin boundary condition in the M-V-D. To overcome this, we use the abstract theory of discretizations based on the notion of stability threshold to analyze the scheme. Stability, and convergence of the proposed numerical scheme are established.