Maximum Principle Preserving Finite Difference Scheme for 1-D Nonlocal-to-Local Diffusion Problems

TL;DR

This paper introduces a finite difference scheme for 1-D nonlocal-to-local diffusion problems that preserves the maximum principle and flux balance at the discrete level, ensuring stability and convergence.

Contribution

It develops a novel finite difference scheme that maintains maximum principle and flux balance for nonlocal-to-local diffusion models, with rigorous proofs and stability analysis.

Findings

The scheme preserves the maximum principle and flux balance.

Theoretical proofs confirm stability and convergence.

Numerical tests validate the scheme's effectiveness.

Abstract

In a recent paper (see [7]), a quasi-nonlocal coupling method was introduced to seamlessly bridge a nonlocal diffusion model with the classical local diffusion counterpart in a one-dimensional space. The proposed coupling framework removes inconsistency on the interface, preserves the balance of fluxes, and satisfies the maximum principle of diffusion problem. However, the numerical scheme proposed in that paper does not maintain all of these properties on a discrete level. In this paper we resolve these issues by proposing a new finite difference scheme that ensures the balance of fluxes and the discrete maximum principle. We rigorously prove these results and provide the stability and convergence analyses accordingly. In addition, we provide the Courant-Friedrichs-Lewy (CFL) condition for the new scheme and test a series of benchmark examples which confirm the theoretical findings.