Analysis of accuracy of a finite volume scheme for diffusion equations on distorted meshes

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Abstract

We investigate the convergence of a finite volume scheme for the approximation of diffusion operators on distorted meshes. The method was originally introduced by Hermeline [F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys. 160 (2000) 481–499], which has the advantage that highly distorted meshes can be used without the numerical results being altered. In this work, we prove that this method is of first-order accuracy on highly distorted meshes. The results are further extended to the diffusion problems with discontinuous coefficient and non-stationary diffusion problems. Numerical experiments are carried out to confirm the theoretical predications.

Introduction

Investigating the numerical schemes with high accuracy for the diffusion equation on distorted meshes is very important in Lagrangian hydrodynamics and magnetohydrodynamics. The finite volume method is a discretization technique for solving partial differential equations (PDEs), which is obtained by integrating the PDE over a control volume, and it represents in general the conservation of a quantity of interest, such as mass, momentum, or energy. Due to this natural association, the finite volume method is widely used in practical problems.

A finite volume scheme solving diffusion equation on non-rectangular meshes is introduced in [6], which is the so-called nine point scheme on arbitrary quadrangles. In [2], [4] a similar scheme for the stationary diffusion equation with smooth coefficient is proposed, but the method of constructing the scheme is different. Moreover, in [2] the unknowns are defined at both the cell-center and the cell-node of primary mesh, and the dual mesh is formed by connecting the center of neighboring primary mesh. Although [2] does not give the theoretical proof, numerical experiments show that this scheme has indeed second accuracy and the numerical solution is independent of the mesh regularity, which demonstrates that the scheme is efficient in solving diffusion equation on Lagrangian meshes.

There are many papers concerning the other numerical schemes for the diffusion equation on distorted meshes, e.g. [1], [5], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. The support-operators method (SOM) in [9], [10], [11], [12], [13], [14] generally has both cell-center and face-center unknowns or has a dense diffusion matrix, and often leads to a symmetric and positive definite coefficient matrix. However, it does not offer an explicit discrete flux expression. The method of multipoint flux approximations (MPFA) [15], [16] whose discrete flux defined on the sub-edge, obtains the explicit discrete flux expression by solving a small scale linear system and often leads to a non-symmetric coefficient matrix for general quadrilateral. The paper [17] presents the relationships among some locally conservative discretization methods.

An interesting application of solving diffusion problems on distorted meshes is the numerical adaptive grid generation (see, e.g. [3], [7], [8], [19]), which is a common tool for use in the numerical solution of partial differential equations on arbitrarily shaped regions. In particular, the meshes are often irregular for solving adaptive moving mesh equations, when the physical governing equations describe certain physical process that undergoes a motion or change with large amplitude.

This paper continues to theoretically analyze the scheme presented in [2], and gives the construction of the scheme for the diffusion equations with discontinuous coefficient. Based on a different approach of devising the finite volume scheme from those in [4], [5], [6], we will show the convergence theorem for the finite volume scheme. Specially, we show that the scheme is first-order convergence. Moreover, the theoretical analysis is extended to non-stationary diffusion problem, and the corresponding convergence theorem is obtained. Numerical experiments will verify the theoretical results.

The rest of the paper is organized as follows. In Section 2, we describe the construction of the finite volume scheme on distorted meshes, and state the convergence theorem. In Section 3, the convergence of the scheme is proved, and in Section 4 the scheme for the diffusion equation with discontinuous coefficient is constructed. Moreover the corresponding results for the non-stationary diffusion equation are given in Section 5. Then some numerical examples are presented to show its performance on several test problems in Section 6. Finally, in Section 7 we end with some concluding remarks.

Section snippets

Construction of finite volume scheme

Consider the following stationary diffusion problem:-·(κu)=finΩ,(κu)·ν+λu=gonΩ,where Ω is an open bounded polygonal set of R2 with boundary ∂Ω.

Suppose the following conditions (H1) are fulfilled:

  • (i)

    There are positive constants κ1, κ2, λ1, λ2 such thatκ1κ=κ(x)κ2,λ1λ=λ(x)λ2,xΩ¯.

  • (ii)

    κ = κ(x), λ = λ(x), f = f(x), g = g(x) are continuous functions on Ω¯.

  • (iii)

    The problem (1), (2) has a unique solution u=u(x)C2(Ω¯).

We use a mesh on Ω (called primary mesh) made up of arbitrary polygons. With each (primary)

Proof of Theorem 1

Using Eqs. (3), (4), (5), we getsPpFps=Ppfp(pΩ),Fps,Ω+SpsΩλu=|SpsΩ|gp(pΩ),sDdΩFds+DdΩλu=Ddfd+|DdΩ|gd.Hence there aresPpF¯ps=-sPpRps|Sps|+Ppfp(pΩ),F¯ps,Ω+|SpsΩ|λpu(p)=SpsΩλ(x)(u(p)-u(x))+|SpsΩ|(gp-Rps,Ω)(pΩ),sDdΩF¯ds+|DdΩ|λdu(d)=-sDdΩRds|Sds|+DdΩλ(x)(u(d)-u(x))+Ddfd+|DdΩ|gd.DenoteGps=F¯ps-Fps,Gds=F¯ds-Fds,Gps,Ω=F¯ps,Ω-Fps,Ω.Then, by using of Eqs. (14), (15), (16) and (20), (21), (22), there aresPpGps=-sPpRps|Sps|(pΩ),Gps,

Problems with discontinuous coefficient

In this section, we construct the scheme for the diffusion equations with discontinuous coefficient. Assume that the primary elements are homogeneous, but the material properties may vary between primary elements. Hence, the dual elements may contain different materials, and we denote it by − and +, respectively (see Fig. 3). For convenience, we introduce some new notations (see Fig. 4) here. Let σ be the dual polygon side Sds = [p, v(p, s)]. Let dd,σ be the distance between d and σ, and dv(d,s),σ

Non-stationary diffusion problem

In this section, we will extend the previous results to the following diffusion problem:ut-·(κ(x,t)u)=f(x,t)inΩ×(0,T],(κ(x,t)u)·ν+λ(x,t)u=g(x,t)onΩ×[0,T],u(x,0)=φ(x)onΩ.The notations have the similar meaning as those in the Section 2. Discretize the time segment by 0 = t0 < T1 <  < TN+1 = T, tn = nΔt. We adopt the implicit scheme for the problem (40), (41), (42) and the same discretization as the Section 2 for the diffusion term, source term and the initial-boundary condition. We can get the following

Numerical results

In [2] some interesting numerical results for both linear and nonlinear parabolic equations are given to show the second-order accuracy and the efficiency of the method. In this section we present the numerical results obtained on different meshes for several test problems. Four different types of meshes are used. One is a uniform mesh for the domain Ω = [0, 1] × [0, 1]. The number of cell center unknowns in the x and y directions are I and J respectively, and therefore the total number of mesh

Conclusions and extensions

In this paper the truncation error of the discrete flux across the cell-sides is derived, and then the corresponding convergence theorem has been obtained, i.e., this method is of first-order convergence on highly distorted meshes. Moreover, the corresponding schemes for the diffusion equation with discontinuous coefficient and the non-stationary diffusion equation are also obtained. Numerical results are presented to demonstrate the performance of these schemes, which show that the method

Acknowledgments

We thank Professor Tao Tang, Xudeng Hang and the two reviewers for their numerous constructive comments and suggestions that helped improving the paper significantly.

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The project is supported by the Special Funds for Major State Basic Research Projects 2005CB321703, the National Nature Science Foundation of China (No. 10476002, 60533020).

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