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

TL;DR

This paper addresses numerical solution of the diffusion-augmented McKendrick–Von Foerster equation with nonlocal nonlinear Robin boundary conditions, specifically $u(0,t)-u_x(0,t)=\int_0^{a_\dagger} B(x,s_2(t))u(x,t)\,dx$. The authors propose an explicit finite difference scheme and analyze it using a nonlinear stability-with-threshold framework since standard linear techniques are inapplicable due to nonlinearity. They prove consistency, stability under a mesh-coupled time step $k=r h^2=\lambda h$ with $\lambda+2r\le1$, and convergence via a Stetter-based abstract discretization theorem, assuming Lipschitz smoothness of $d,B,\psi_i$ and smooth solutions. Numerical experiments in bounded domains corroborate the theory, including cases with nonconstant $d,B$ and nonhomogeneous Dirichlet boundaries, demonstrating convergence as the mesh is refined and confirming practical viability for age-structured diffusion models.

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.

Figures (3)

• Figure 1: The exact solution to \ref{['e1']}, and the approximate solutions using \ref{['scheme']} with $d(x,s)$, $B(x,s)$ given in Example \ref{['example1']}; Left: $u(x,0.2)$ (solid line), $\boldsymbol{U}_{0.05}$ (dotted line), $\boldsymbol{U}_{0.01}$ (dash-dotted line), $\boldsymbol{U}_{0.005}$ (dashed line) for $0\leq x \leq 1$, Right: $\mid u(x,0.2)- \boldsymbol{U}_{0.05}\mid$ (dotted line), $\mid u(x,0.2)- \boldsymbol{U}_{0.01}\mid$ (dash-dotted line) and $\mid u(x,0.2)- \boldsymbol{U}_{0.005}\mid$ (dashed line).
• Figure 2: The approximate solutions to \ref{['scheme']} with $d(x,s)$, $B(x,s)$ given in Example \ref{['example2']}; Left: $\boldsymbol{U}_{0.005}$ (solid line), $\boldsymbol{U}_{0.01}$ (dashed line), $\boldsymbol{U}_{0.05}$ (dash-dotted line), $\boldsymbol{U}_{0.1}$ (dotted line) for $0\leq x \leq 1$, Right: $\mid \boldsymbol{U}_{0.005}- \boldsymbol{U}_{0.01}\mid$ (dashed line), $\mid \boldsymbol{U}_{0.005}- \boldsymbol{U}_{0.05}\mid$ (dash-dotted line), and $\mid \boldsymbol{U}_{0.005}- \boldsymbol{U}_{0.1}\mid$ (dotted line).
• Figure 3: The exact solution to \ref{['non_e1']} and the approximate solutions to \ref{['new_scheme']} with $d(x,s)$, $B(x,s)$ given in Example \ref{['example3']}; Left: $u(x,0.2)$ (solid line), $\boldsymbol{U}_{0.05}$ (dotted line), $\boldsymbol{U}_{0.01}$ (dash-dotted line), $\boldsymbol{U}_{0.005}$ (dashed line) for $0\leq x \leq 1$, Right: $\mid u(x,0.2)- \boldsymbol{U}_{0.05}\mid$ (dotted line), $\mid u(x,0.2)- \boldsymbol{U}_{0.01}\mid$ (dash-dotted line) and $\mid u(x,0.2)- \boldsymbol{U}_{0.005}\mid$ (dashed line).

Theorems & Definitions (15)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1
• Theorem 3.1: Consistency
• proof
• Lemma 3.1
• proof
• Theorem 3.2: Stability
• proof