Fourth-order compact finite difference method for solving two-dimensional convection\u2013diffusion equation

Lingyu Li,Zhe Yin,Ziwen Jiang

doi:10.1186/s13662-018-1652-5

Abstract

A fourth-order compact finite difference scheme of the two-dimensional convection–diffusion equation is proposed to solve groundwater pollution problems. A suitable scheme is constructed to simulate the law of movement of pollutants in the medium, which is spatially fourth-order accurate and temporally second-order accurate. The matrix form and solving methods for the linear system of equations are discussed. The theoretical analysis of unconditionally stable character of the scheme is verified by the Fourier amplification factor method. Numerical experiments are given to demonstrate the efficiency and accuracy of the scheme proposed, and these show excellent agreement with the exact solution.

Highlights

In recent years, more and more attention has been paid to the movement of pollutants in groundwater by mathematical modeling [1]
Convection–diffusion equation is a class of very important equations, it can describe many physical phenomena, such as atmospheric pollutants, distribution and diffusion of the oceans and rivers, heat conduction and so many other physical problems even including bacterial concentration
The paper is organized as follows: in Sect. 2, we present a fourth-order compact difference scheme, in which the Crank–Nicolson scheme is used for temporal discretization and a fourth-order compact finite difference scheme dealing with a one-dimensional convection–diffusion equation is applied to the spatial discretization

Summary

Introduction

More and more attention has been paid to the movement of pollutants in groundwater by mathematical modeling [1]. The prediction and evaluation of groundwater dynamic movement and solute transport are important tasks for agricultural pollution and groundwater development [2]. A large number of mathematical models and a variety of effective numerical methods have been widely used to simulate the movement of contaminated groundwater. From the existing research results, we could only get the analytical solutions of a few classic models. In the process of dealing with practical problems, for many mathematical models, especially partial differential equations, their analytical solutions are not available in general. Research for the numerical solutions of partial differential equations is very necessary [3]

Objectives

Conclusion