An extremum-preserving finite volume scheme for convection-diffusion equation on general meshes

Abstract

We present an extremum-preserving finite volume scheme for the convection-diffusion equation on general meshes in this article. The harmonic averaging point locating at the interface of heterogeneity are utilized to define the auxiliary unknowns. The second-order upwind method with a slope limiter is used for the discretization of convection flux. This scheme has only cell-centered unknowns and possesses a small stencil. The extremum-preserving property of this scheme is proved by standard assumption. Numerical results demonstrate that the extremum-preserving scheme is an efficient method in solving the convection-diffusion equation on distorted meshes.

Introduction

This paper is contributed to present an extremum-preserving finite volume scheme for the convection-diffusion problem

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \text{div}(\upsilon \mathrm{u}-\Lambda \nabla \mathrm{u})=\mathrm{f}\text{in}\Omega ,}\hfill \\ & {\displaystyle \mathrm{u}=\mathrm{g}\text{on}\partial \Omega ,}\hfill \end{array}\end{array}$$

where Ω is an open bounded connected general subset of ${\mathbb{R}}^d(d=2\text{or}3),$ ∂Ω is the boundary of Ω, Λ is a symmetric positive definite d × d diffusion tensor, g ∈ L^∞(∂Ω) is the boundary condition, $\upsilon \in C^1\left(\overline{\Omega }\right){}^2$ is the velocity field, $\text{div}\upsilon \ge 0,$ and f ∈ L²(Ω) is source term, respectively.

The convection-diffusion equation has many applications, for example the gas dynamics, the process of groundwater transport of a solute in porous media [1], [2], [3]. One important property of the solution for convection-diffusion equation is the extremum principle. The physical quantities concentration and temperature without input will satisfy the extremum principle. And, the numerical solutions also should conform with this property. Therefore, a reliable extremum-preserving scheme is needed for the numerical simulation of convection-diffusion equation.

Some effective numerical schemes have been developed for the convection-diffusion equation to solve the convection-dominated problem [4], [5], [6], [7], [8]. These schemes only can be used on the regular meshes and don’t satisfy the discrete extremum principle. Recently, some finite volume schemes are presented [9], [10], [11]. In [11], the second-order upwind method with slope limiter is first be used to discretize the convection flux. Then, several monotone algorithms [12], [13] of the convection-diffusion equation been proposed based on the second-order upwind method with slope limiter. The schemes proposed in [11], [14] are extremum-preserving, but only considered the case under geometric restrictions. Besides, the inverse distance weighting method is used as the interpolation algorithm of [14]. It is well known that this interpolation method has large accuracy loss on distorted meshes. Thus, few numerical scheme can solve the convection-diffusion equation extremum-preserving and second-order accurate on the 2D or 3D distorted meshes.

In this article, we will develop the scheme presented in [9], [11], [15] to construct a new extremum-preserving finite volume scheme for convection-diffusion equation. The nonlinear extremum-preserving scheme in [15] is used to discretize the diffusion flux, and the method of discretize the convection flux presented in [9], [11] is also applied. In summary, this new extremum-preserving scheme has the following characteristics:

• it is locally conservative;

• it satisfies discrete extremum principle;

• it allows anisotropic diffusion tensor;

• it is reliable on unstructured meshes that may be highly distorted;

• it has second-order accuracy for the smooth solutions.

In addition, this scheme has only the cell center unknown. By using the harmonic averaging points as auxiliary unknowns, the interpolation procedure on the distorted meshes will be simplified. This property makes our scheme easy to implement in solving the convection-diffusion equation on the distorted polygonal meshes and polyhedral meshes.

The rest part of this paper is organized as follows. In Section 2, the extremum-preserving finite volume scheme is constructed. Then in Section 3, we present some 2D and 3D numerical examples to illustrate the accuracy of the scheme. Finally, some conclusions are given in Section 4.