Complete Flux Scheme for Elliptic Singularly Perturbed Differential-Difference Equations

TL;DR

This paper introduces a novel complete flux scheme based on finite volume methods for solving elliptic singularly perturbed differential-difference equations, demonstrating stability and convergence through theoretical analysis and numerical tests.

Contribution

The paper presents a new complete flux scheme specifically designed for elliptic SPDDEs, including stability, consistency, and convergence proofs.

Findings

Scheme is stable and convergent for test problems

Effective in handling singular perturbations

Provides accurate numerical solutions

Abstract

In this study, we propose a new scheme named as complete flux scheme (CFS) based on the finite volume method for solving singularly perturbed differential-difference equations (SPDDEs) of elliptic type. An alternate integral representation for the flux is obtained which plays an important role in the derivation of CF scheme. We have established the stability, consistency and quadrature convergence of the proposed scheme. The scheme is successfully implemented on test problems.