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

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.)en, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the celledge 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.


Introduction
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. e extremum principle is an important property of solutions for the convection-diffusion equation. It includes minimum principle and maximum principle. e authors of [1,2] pointed out that the discrete maximum principle (DMP) plays an important role in proving the existence and uniqueness of discrete solution, enforcing numerical stability, and deriving convergence for a sequence of approximate solutions [3]. Pert [4] pointed out that a scheme violating extremum principle can lead to two problems: fully implicit discretization with large time-steps has relatively poor accuracy, and spurious negative values are generated. Moreover, 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.
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 convectiondiffusion equation, which says that it can only guarantee nonnegative bound of the numerical solution. Sheng and Yuan [2] pointed out that the scheme without positivitypreserving can lead to the violation of the entropy constraints of the second law of thermodynamics, causing heat to flow from regions of lower temperature to higher temperature. In regions of large temperature variations, this can cause the temperature to become negative. e 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 [5][6][7][8]. Some nonlinear methods have been developed [9][10][11][12][13][14][15][16][17][18][19][20][21] for general diffusion or convection-diffusion equations, which guarantee the positivity on general or distorted meshes for general tensor coefficients.
Bertolazzi and Manzini [22] proposed a MUSCL-like cell-centered finite volume method, where the discretization of advective fluxes is based on a least-square reconstruction of the vertex values from cell averages. Lipnikov et al. [23] proposed a new slope limiting technique based on a specially minimal nonlinear correction, which follows the ideas of the monotonic upstream-centered scheme for conservation laws (MUSCL). en, in the studies by Wang et al. and Zhang et al. [16,24], the limiting technique is used to avoid nonphysical oscillation. 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.
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 [18] and convective flux [21], the adaptive approach of choosing stencil is applied. Positivitypreserving scheme can only guarantee nonnegative bound of the numerical solution. Considering that the computation of value on the cell edge and the value of cell-centered unknowns may be out of bound, a correcting technique is introduced. Our scheme is constructed by a nonlinear combination technique and has second-order accuracy for the solution and first-order for the flux. e article is organized as follows. e model problem is described, and some notations are introduced in Section 2. e main process of construction for the 2D steady convectiondiffusion equation is given in Section 3. In Section 4, several numerical tests are exhibited to illustrate the features of our scheme. At last, some conclusions are given in Section 5.

The Problem and Notation
Consider the following stationary convection-diffusion problem for unknown function u � u(x): where Ω is a bounded polygonal domain in is a velocity vector field. Assume that the functions v → (x), f(x), and g(x) satisfy the constraints listed as follows: and there are two positive constants λ 1 and λ 2 such that e solvability of the problem (1)-(2) has been given, and the maximum and minimum principle can be found in the study by Gilbarg and Trudinger [25].
We use a mesh on Ω made up of arbitrary convex polygon cells.
e set of all cells, edges, and nodes are denoted by T, ε, and N, respectively.
We denote the cell by K, L, . . . , and the cell center is also denoted by K, L, . . . ,. In addition, the common edge of two cells K and L is denoted by σ, i.e., σ � K|L ∈ Ε. e cell-edge σ is also denoted by AB, and the midpoint of σ is denoted by M. Moreover, we denote P 1 and P 2 are two adjacent midpoints of cell K (Figure 1).
Let n → K,σ (or n → L,σ ) be the unit outer normal vector on the cell-edge σ of cell K (or L), κ T be the transpose of matrix κ, and E K be the set of all edges of cell K. Denote ε int � ε ∩ Ω and ε ext � ε ∩ zΩ. Denote h � (sup K∈T m(K)) 1/2 , where m(K) is the area of cell K.
Integrating (1) over the cell K, we obtain where the diffusive and convective flux are defined as

e Diffusive Flux.
Following the idea in the study by Sheng and Yuan [18], the adaptive approach of choosing stencil is applied for the approximation of the diffusive flux (equation (6)) together with a nonlinear combination technique. In the method, the two nonnegative parameters are introduced to define a nonlinear two-point flux. en, the continuity of normal flux on the cell edge is used to give the final discretization of the diffusive flux. At last, the continuity of normal flux is used to obtain the value of u M . First, we give a brief review of the construction. Figure 1 shows that a ray originating at the point K along the direction κ T n → K,σ must intersect one segment connecting two neighboring midpoints of edge of cell K, where the two midpoints are denoted by P 1 and P 2 , and the cross point is denoted by O 1 . Similarly, a ray originating at the midpoint M along the direction − κ T n → K,σ must intersect one certain KP 4 , where P 4 must be one vertex of σ, and the cross point is 4 be some unit tangential vectors along their corresponding directions, respectively. θ i (i � 1, . . . , 4) are some corresponding angles. Hence, we established the following relations: Mathematical Problems in Engineering Substituting equation (8) into equation (6) and neglecting the high-order terms, we have where Similarly, substituting equation (9) into equation (6), we have where Combined with equations (10)- (12), the discrete normal flux on σ can be defined as follows: where μ 1 and μ 2 are some nonlinear coefficients with μ 1 + μ 2 � 1, which will be given later.
In order to assure μ 1 and μ 2 are positive, two additional parameters ω 1 and ω 2 are introduced later. According to the different positions of P 1 and P 2 , three cases exist.
We assume that u P 1 ≠ u M and u P 2 ≠ u M , the normal flux (14) can be rewritten as In order to obtain the two-point flux approximation, the last two terms of the above expression should vanish; hence, μ 1 and μ 2 are given as follows: Hence, (15) can be expressed as follows: where In order to assure μ 1 > 0 and μ 2 > 0, two parameters ω 1 and ω 2 can be chosen such that If (14) can be expressed in the similar form (17) by using the above method.
Similarly, on the edge σ of the cell L, we have Using the continuity of normal flux F K,σ + F L,σ � 0 on edge σ, we obtain Substitute equation (22) into equation (17) to obtain the nonlinear two-point diffusive flux on σ � K|L: where A K,σ � A K,σ,1 A L,σ,2 /A K,σ,2 + A L,σ,2 , and A L,σ � A K,σ,2 A L,σ,1 /A K,σ,2 + A L,σ,2 . From the computation of vertex unknowns, a method with second-order accuracy has been proposed in the study by Sheng and Yuan [26]. We know that u M > 0 as long as u K > 0 and u L > 0 in equation (22).

e Convective Flux.
We focus on the expression of convective flux in equation (7) for ∀σ � K|L ∈ ε int and obtain where u M is the value of midpoint M on the cell-edge σ, and v + K,σ � In order to ensure that the discretization of equation (24) has the same structure as the scheme (23), we divide the integral term into positive part (v + K,σ ) and negative part (v − K,σ ). Moreover, the property of upwind is also considered. Neglecting the high-order terms, we have the following approximate expression of the upwind formula [27]: In order to approximate the continuous flux G K,σ on the cell-edge σ with second-order accuracy, we propose the following method.
A local stencil is given in Figure 2. For the cell K, M is the midpoint of an arbitrary edge and M 1 , M 2 are the other two midpoints adjacent to it. We denote S △MM 1 K be the area of triangle MM 1 K and define where . For a special case (Figure 3), we set the cell K as a triangle and define It is obvious that u K is a second-order approximation to u K , i.e., |u(K) − M∈σ,σ∈E K en, the approximation of G K,σ on cell-edge σ can be defined as follows.
For σ ∈ E int , we define where For σ ∈ E ext , we define where 4 Mathematical Problems in Engineering

e Finite Volume Scheme and Picard Iteration.
By using the definition of discretization of diffusive and convective flux, the finite volume scheme can be constructed as follows: where f K � f(K) and g M i � g(x M i ).
Let U be the discrete unknown vector and A(U) be the coefficient matrix. A nonlinear algebraic system of the schemes (33) and (34) can be formed: A(U)U � F. e A(U) is assembled by the coefficients of diffusive term F K,σ and convective term G K,σ . We use the Picard nonlinear iteration method to solve the system: choose a small value E non > 0 and initial vector U 0 > 0 and repeat for nonlinear iteration index s � 1, 2, . . . ,: (1) Solve A(U (s− 1) )U (s) � F (2) Stop if ‖A(U (s) )U (s) − F‖ 2 ≤ Ε non ‖A(U (0) )U (0) − F‖ 2 e linear algebraic system with coefficient matrix A(U (s− 1) ) is solved by the biconjugate gradient stabilized (BiCGSTab) method, and the linear iterations are terminated when relative norm of the initial residual becomes smaller than ε lin .

e Algorithm.
In this subsection, we describe the detailed algorithm.

Remark 2.
e value of ∀u (s) M can be obtained by equation (22). If ∀u (s) where the common edge σ of two cells K and L is also denoted by AB (Figure 1 (35) e detailed proof of positivity is given in the study by Yuan and Sheng [11]. Now, we state our conclusion, which says that our scheme can avoid the numerical solution beyond the upper bound. Denote u max � max 0, u K , u M , ∀K ∈ T, ∀M ∈ ε .

Numerical Experiments
In order to demonstrate the accuracy and robustness of the scheme, we test several problems and take ε non � 1.0e − 6 and Mathematical Problems in Engineering ε lin � 1.0e − 10 . e convergence order can be obtained by the following formula: where N 1 and N 2 represent different number of cells, and Error(N 1 ) and Error(N 2 ) are the corresponding L 2 errors.

e Problem with Anisotropic Diffusion Tensor.
Consider the problems (1) and (2) and the diffusion coefficient is First, we test the accuracy of our scheme on random quadrilateral meshes shown in Figure 4. Table 1 gives the corresponding L 2 error and the numbers of nonlinear iteration numbers it # non with a different parameter ε. We can see that our scheme obtains second-order accuracy for the solution and at least first-order accuracy for the flux. e average number of nonlinear iterations is 36.8 when ε � 10 − 6 . However, for ε � 1.0, the corresponding number increases while the number of cell increases.

e Problem with Discontinuous
and the diffusion coefficient is where c 0 � 40. We test this problem on random triangle meshes shown in Figure 5. e numerical results with a different parameter ε � 1.0, 10 − 6 are given in Table 2. We can see that our scheme almost obtain second-order accuracy for the solution and at least first-order accuracy for the flux. e average numbers of nonlinear iterations are 25 and 23 for ε � 1.0 and ε � 1.0, 10 − 6 , respectively. en, in order to illustrate the efficiency of our scheme, the comparison of accuracy between the studies by Lan et al. and Zhang et al. [19,24] and present scheme is given in Table 3. As can be seen from the table, the accuracy is 1.5 in the study by Zhang et al. [24] and more than 2.0 in the study by Lan et al. [19] and our new scheme. However, the computed results in our scheme can approximate the exact solution more well.
We take k 1 � 1, k 2 � 100, and θ � 5π/6. e analytical solution u(x, y) is unknown, but the minimum principle states that it is nonnegative. It is a challenging task to solve it accurately because they can result in significant violation of the positivity and even produce a numerical solution with nonphysical oscillations. We will show that our new nonlinear scheme can also obtain the nonnegative solution.
First, we test the new scheme on the random quadrilateral meshes with 128 × 128 cells. e corresponding distribution is similarly shown in Figure 6), and the numerical solution is given in Figure 7. e minimum value is u min � 0 and the maximum value is u max � 6.7603 × 10 − 4 , which show that our scheme preserves the positivity of the solution and does not produce any nonphysical oscillations. en, we test it on the random triangular meshes. e computed results show that u min � 0 and u max � 6.7582 × 10 − 4 . ese computed results illustrate that our new scheme preserves the positivity of numerical solutions and satisfies the discrete minimum principle.

Nonphysical Oscillations.
We also consider the last nonsmooth anisotropic solution and compute it on the random quadrilateral meshes. Here, we reset f � 0. e computational domain is a unit square with a hole, Ω � [0, 1] 2 ∖[4/9, 5/9] 2 , so that the boundary zΩ is composed of two disjoint parts Γ 0 and Γ 1 as shown in Figure 8 where the number of cell is 72 × 72. Γ 0 is the exterior boundary, and Γ 1 is the interior boundary. We set g � 0 on Γ 0 and g � 2 on Γ 1 . e numerical solutions on the random quadrilateral meshes are shown in Figure 9. e computed results show that u min � 6.94E − 13 and u max � 1.98. It means that the minimum 0 is attained on the Γ 0 , and the maximum 2 is attained on the Γ 1 . So, these computed results illustrate that our scheme can avoid the numerical solution beyond the upper bound and does not produce any nonphysical oscillations.
en, the computed results without the correct method in Remarks 1 and 2 are shown in Figure 10, and u min � 3.87E − 6 and u max � 1.91. However, the numerical oscillations are produced in the computational domain.   Figure 5: e random triangle meshes.   Mathematical Problems in Engineering 9

Conclusion
e 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. en, 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.
Data Availability e authors confirm that the data supporting the findings of this study are available within the article (No. 7343716).

Conflicts of Interest
e authors declare that they have no conflicts of interest.