Nonlinear Finite Volume Scheme Preserving Positivity for 2D Convection-Diffusion Equations on Polygonal Meshes

Bin Lan,Jianqiang Dong

doi:10.1155/2020/7343716

Abstract

In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.

Highlights

Convection-diffusion equations are widely used in the fields of solid mechanics, material science, image processing, and so on
An accurate numerical method must maintain the fundamental properties of practical problems. e extremum principle is an important property of solutions for the convection-diffusion equation
It is proved that a linear operator, resulting from the discretization of diffusion equations, satisfies extremum principle if and only if it is both differential and nonnegativity maintaining

Summary

Introduction

Convection-diffusion equations are widely used in the fields of solid mechanics, material science, image processing, and so on It is both theoretically and practically important to investigate numerical methods for such equations. E extremum principle is an important property of solutions for the convection-diffusion equation. Positivity-preserving is one of the key requirements to discrete schemes for the convectiondiffusion equation, which says that it can only guarantee nonnegative bound of the numerical solution. In the study by Lan et al [21], a new upwind scheme is used to discretize the convective flux, and the method did not introduce any slope limiting technique. We develop a nonlinear FV scheme, which satisfies DEP for convection-diffusion problems on arbitrary convex polygonal meshes.

The Problem and Notation

Construction of the Scheme

Numerical Experiments