Monotone Difference Schemes for Convection-Dominated Diffusion-Reaction Equations Based on Quadratic Spline

TL;DR

This paper introduces a monotone difference scheme based on quadratic splines for efficiently solving one-dimensional convection-diffusion-reaction equations with variable coefficients, especially effective when convection dominates diffusion.

Contribution

The paper presents a novel three-point monotone difference scheme utilizing parabolic splines that accurately reproduces solutions and is highly effective for convection-dominated problems.

Findings

High accuracy in reproducing solutions with dominant convection

Effective handling of small diffusion parameters

Numerical tests confirm scheme's efficiency and stability

Abstract

A three-point monotone difference scheme is proposed for solving a one-dimensional non-stationary convection-diffusion-reaction equation with variable coefficients. The scheme is based on a parabolic spline and allows to linearly reproduce the numerical solution of the boundary value problem over the integral segment in the form of the function which continuous with its first derivative. The constructed difference scheme give a highly effective tool for solving problems with a small parameter at the older derivative in a wide range of output data of the problem. In the test case, numerical and exact solutions of the problem are compared with the significant dominance of the convective term of the equation over the diffusion. Numerous calculations showed the high efficiency of the new monotonous scheme developed.