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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.

Convection-diffusion equations are widely used in the fields of solid mechanics, material science, image processing, and so on. So, it is both theoretically and practically important to investigate numerical methods for such equations. An accurate numerical method must maintain the fundamental properties of practical problems. The extremum principle is an important property of solutions for the convection-diffusion equation. It includes minimum principle and maximum principle. The authors of [

In general cases, the discrete extremum principle (DEP) is more restrictive than monotonicity (positivity-preserving). However, it is difficult to construct a reliable discretization method that satisfies the DEP on arbitrary convex polygonal meshes. Hence, positivity-preserving is one of the key requirements to discrete schemes for the convection-diffusion equation, which says that it can only guarantee nonnegative bound of the numerical solution. Sheng and Yuan [

The finite volume methods (FVM) guarantee the local conservation. But many classical schemes fail to maintain positivity for strong anisotropic diffusion tensors or on distorted meshes [

Bertolazzi and Manzini [

In this paper, we develop a nonlinear FV scheme, which satisfies DEP for convection-diffusion problems on arbitrary convex polygonal meshes. Following the idea of the discretization for diffusive flux [

The article is organized as follows. The model problem is described, and some notations are introduced in Section

Consider the following stationary convection-diffusion problem for unknown function

Assume that the functions

The solvability of the problem (

We use a mesh on

We denote the cell by

Local stencil 1.

Let

Integrating (

Following the idea in the study by Sheng and Yuan [

Figure

Let

Substituting equation (

Similarly, substituting equation (

Combined with equations (

In order to assure

We assume that

In order to obtain the two-point flux approximation, the last two terms of the above expression should vanish; hence,

Hence, (

In order to assure

If

We let

If

Similarly, on the edge

Using the continuity of normal flux

Substitute equation (

From the computation of vertex unknowns, a method with second-order accuracy has been proposed in the study by Sheng and Yuan [

We focus on the expression of convective flux in equation (

In order to ensure that the discretization of equation (

Neglecting the high-order terms, we have the following approximate expression of the upwind formula [

In order to approximate the continuous flux

A local stencil is given in Figure

Local stencil 2.

Local stencil of triangle.

It is obvious that

Then, the approximation of

For

For

By using the definition of discretization of diffusive and convective flux, the finite volume scheme can be constructed as follows:

Let

Solve

Stop if

The linear algebraic system with coefficient matrix

In this subsection, we describe the detailed algorithm.

Step 1. Initialize

Step 2. When

compute

compute initial residual

Step 3. When

solve

correct

compute

compute

compute residual

whether

Step 4. Stop.

For

The value of

It should be noted that the algorithm in Remark

Let

The detailed proof of positivity is given in the study by Yuan and Sheng [

Now, we state our conclusion, which says that our scheme can avoid the numerical solution beyond the upper bound. Denote

We assume

In order to demonstrate the accuracy and robustness of the scheme, we test several problems and take

Consider the problems (

First, we test the accuracy of our scheme on random quadrilateral meshes shown in Figure

The random quadrilateral meshes.

Accuracy for the problem with anisotropic coefficient.

Cells | 144 | 576 | 2304 | 9216 | 36864 | |
---|---|---|---|---|---|---|

— | 2.05 | 1.84 | 1.94 | 1.94 | ||

— | 2.07 | 1.54 | 2.03 | 0.99 | ||

55 | 85 | 116 | 126 | 153 | ||

— | 2.09 | 1.96 | 1.98 | 1.92 | ||

— | 2.07 | 1.54 | 2.03 | 0.99 | ||

36 | 40 | 36 | 33 | 39 |

Consider the problems (

We test this problem on random triangle meshes shown in Figure

The random triangle meshes.

Accuracy for the problem with discontinuous coefficient.

Cells | 288 | 1152 | 4608 | 18432 | 73728 | |
---|---|---|---|---|---|---|

1.0 | ||||||

— | 1.81 | 1.86 | 2.14 | 2.03 | ||

— | 2.07 | 1.54 | 2.03 | 0.99 | ||

23 | 24 | 26 | 26 | 26 | ||

— | 1.94 | 1.91 | 1.98 | 1.98 | ||

— | 2.07 | 1.54 | 2.03 | 0.99 | ||

23 | 24 | 23 | 23 | 23 |

Then, in order to illustrate the efficiency of our scheme, the comparison of accuracy between the studies by Lan et al. and Zhang et al. [

Comparison of accuracy with discontinuous coefficient

Method | Mesh | Cells | 256 | 1024 | 4096 | 16384 |
---|---|---|---|---|---|---|

Zhang et al. [ | Uniform | 1.10 | 3.98 | 1.42 | 5.02 | |

Order | — | 1.47 | 1.49 | 1.50 | ||

Random | 1.10 | 3.95 | 1.38 | 5.11 | ||

Order | — | 1.48 | 1.52 | 1.43 | ||

Lan et al. [ | Uniform | 4.23 | 9.77 | 2.31 | 5.50 | |

Order | — | 2.12 | 2.08 | 2.07 | ||

Random | 6.87 | 1.57 | 3.78 | 8.96 | ||

Order | — | 2.13 | 2.05 | 2.08 | ||

Present | Uniform | 9.34 | 1.82 | 3.63 | 8.43 | |

Order | — | 2.36 | 2.32 | 2.11 | ||

Random | 9.16 | 1.89 | 4.04 | 9.85 | ||

Order | — | 2.28 | 2.23 | 2.04 |

Now, we consider the problem (

We take

The analytical solution

First, we test the new scheme on the random quadrilateral meshes with

The random quadrilateral meshes.

The numerical solutions on the random quadrilateral meshes.

These computed results illustrate that our new scheme preserves the positivity of numerical solutions and satisfies the discrete minimum principle.

We also consider the last nonsmooth anisotropic solution and compute it on the random quadrilateral meshes. Here, we reset

The random quadrilateral meshes with a hole.

The numerical solutions on the random quadrilateral meshes are shown in Figure

The numerical results on the random quadrilateral meshes.

Then, the computed results without the correct method in Remarks

The numerical results without corrections on the random quadrilateral meshes.

The aim of this paper is to build a nonlinear finite volume scheme preserving positivity for solving the 2D convection-diffusion equation on arbitrary convex polygonal meshes. We first develop the nonlinear positive finite volume scheme. Then, a corrected method is proposed, and the numerical solution beyond the upper bound can be avoided.

Our scheme includes only cell-centered unknowns. Numerical results show that our new scheme obtains second-order accuracy for the solution and first-order accuracy for the flux. In addition, it can not only keep the positivity but also do not produce any oscillation.

The authors confirm that the data supporting the findings of this study are available within the article (No. 7343716).

The authors declare that they have no conflicts of interest.

This work was partially supported by the Natural Science Foundation of Ningxia (2020AAC03235); General School Level Project (12020000153); National Natural Science Foundation of China (11601013 and 11772165); First-Class Disciplines Foundation of Ningxia (NXYLXK2017B09); and Natural Science Foundation of Ningxia (2020AAC03233).