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.
1. 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. The extremum principle is an important property of solutions for the convection-diffusion equation. It includes minimum principle and maximum principle. The 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 convection-diffusion equation, which says that it can only guarantee nonnegative bound of the numerical solution. Sheng and Yuan [2] pointed out that the scheme without positivity-preserving 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.
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 [5–8]. Some nonlinear methods have been developed [9–21] 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). Then, 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. Positivity-preserving 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.
The article is organized as follows. The model problem is described, and some notations are introduced in Section 2. The main process of construction for the 2D steady convection-diffusion 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.
2. The Problem and Notation
Consider the following stationary convection-diffusion problem for unknown function :where is a bounded polygonal domain in with boundary , is a known diffusive coefficient, and is a velocity vector field.
Assume that the functions , and satisfy the constraints listed as follows:and there are two positive constants and such that
The 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 made up of arbitrary convex polygon cells. The set of all cells, edges, and nodes are denoted by , and , respectively.
We denote the cell by and the cell center is also denoted by . In addition, the common edge of two cells and is denoted by , i.e., . The cell-edge is also denoted by , and the midpoint of is denoted by . Moreover, we denote and are two adjacent midpoints of cell (Figure 1).
Let (or ) be the unit outer normal vector on the cell-edge of cell (or ), be the transpose of matrix , and be the set of all edges of cell . Denote and . Denote , where is the area of cell .
Integrating (1) over the cell , we obtainwhere the diffusive and convective flux are defined as
3. Construction of the Scheme
3.1. The 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. Then, 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 . First, we give a brief review of the construction.
Figure 1 shows that a ray originating at the point along the direction must intersect one segment connecting two neighboring midpoints of edge of cell , where the two midpoints are denoted by and , and the cross point is denoted by . Similarly, a ray originating at the midpoint along the direction must intersect one certain , where must be one vertex of , and the cross point is denoted by .
Let , and be some unit tangential vectors along their corresponding directions, respectively. are some corresponding angles. Hence, we established the following relations:
Substituting equation (8) into equation (6) and neglecting the high-order terms, we havewhere
Similarly, substituting equation (9) into equation (6), we havewhere
Combined with equations (10)–(12), the discrete normal flux on can be defined as follows:where and are some nonlinear coefficients with , which will be given later.
In order to assure and are positive, two additional parameters and are introduced later. According to the different positions of and , three cases exist.
We assume that and , 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, and are given as follows:
Hence, (15) can be expressed as follows:where
In order to assure and , two parameters and can be chosen such that
If
We let .
If or , equation (14) can be expressed in the similar form (17) by using the above method.
Similarly, on the edge of the cell , we have
Using the continuity of normal flux on edge , we obtain
Substitute equation (22) into equation (17) to obtain the nonlinear two-point diffusive flux on :where , and .
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 as long as and in equation (22).
3.2. The Convective Flux
We focus on the expression of convective flux in equation (7) for and obtainwhere is the value of midpoint on the cell-edge , and
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 and negative part . 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 on the cell-edge with second-order accuracy, we propose the following method.
A local stencil is given in Figure 2. For the cell , is the midpoint of an arbitrary edge and are the other two midpoints adjacent to it. We denote be the area of triangle and definewhere . For a special case (Figure 3), we set the cell as a triangle and define , where
It is obvious that is a second-order approximation to , i.e., if the solution .
Then, the approximation of on cell-edge can be defined as follows.
For , we definewhere
For , we definewhere
3.3. The 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 and .
Let be the discrete unknown vector and be the coefficient matrix. A nonlinear algebraic system of the schemes (33) and (34) can be formed: . The is assembled by the coefficients of diffusive term and convective term . We use the Picard nonlinear iteration method to solve the system: choose a small value and initial vector and repeat for nonlinear iteration index :(1)Solve (2)Stop if
The linear algebraic system with coefficient matrix 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 .
3.4. The Algorithm
In this subsection, we describe the detailed algorithm. Step 1. Initialize , , and . Step 2. When ,(a)compute and ;(b)compute initial residual . Step 3. When (a)solve ;(b)correct , see Remark 1;(c)compute and correct , , see Remark 2;(d)compute and ;(e)compute residual ;(f)whether , if true, then go to (Step 4), otherwise, go to (Step 3). Step 4. Stop.
Remark 1. For , if , let .
Remark 2. The value of can be obtained by equation (22). If , where the common edge of two cells and is also denoted by (Figure 1). Let .
It should be noted that the algorithm in Remark 1 is important to avoid the numerical solution beyond the upper bound where the numerical results need to depend on nonnegative initial values of nonlinear iteration.
Theorem 1. Let , and linear systems in Picard iterations are solved exactly. Then,
The 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 .
We assume . Using Remark 1, we know that .
4. Numerical Experiments
In order to demonstrate the accuracy and robustness of the scheme, we test several problems and take and . The convergence order can be obtained by the following formula:where and represent different number of cells, and and are the corresponding errors.
4.1. The Problem with Anisotropic Diffusion Tensor
Consider the problems (1) and (2) with Dirichlet boundary condition on , and take . The exact solution isand 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 error and the numbers of nonlinear iteration numbers with a different parameter . We can see that our scheme obtains second-order accuracy for the solution and at least first-order accuracy for the flux. The average number of nonlinear iterations is 36.8 when . However, for , the corresponding number increases while the number of cell increases.
4.2. The Problem with Discontinuous Coefficient
Consider the problems (1) and (2) with Dirichlet boundary condition on , and take . The exact solution isand the diffusion coefficient iswhere .
We test this problem on random triangle meshes shown in Figure 5. The numerical results with a different parameter 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. The average numbers of nonlinear iterations are 25 and 23 for and , respectively.
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. [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.
4.3. Positivity of Numerical Solutions
Now, we consider the problem (1)-(2) in the unit square with homogeneous Dirichlet boundary conditions. Take , and set
We take , and .
The analytical solution 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 cells. The corresponding distribution is similarly shown in Figure 6), and the numerical solution is given in Figure 7. The minimum value is and the maximum value is , which show that our scheme preserves the positivity of the solution and does not produce any nonphysical oscillations. Then, we test it on the random triangular meshes. The computed results show that and .
(a)
(b)
These computed results illustrate that our new scheme preserves the positivity of numerical solutions and satisfies the discrete minimum principle.
4.4. Nonphysical Oscillations
We also consider the last nonsmooth anisotropic solution and compute it on the random quadrilateral meshes. Here, we reset . The computational domain is a unit square with a hole, , so that the boundary is composed of two disjoint parts and as shown in Figure 8 where the number of cell is . is the exterior boundary, and is the interior boundary. We set on and on .
The numerical solutions on the random quadrilateral meshes are shown in Figure 9. The computed results show that and . It means that the minimum 0 is attained on the , and the maximum 2 is attained on the . So, these computed results illustrate that our scheme can avoid the numerical solution beyond the upper bound and does not produce any nonphysical oscillations.
(a)
(b)
Then, the computed results without the correct method in Remarks 1 and 2 are shown in Figure 10, and and . However, the numerical oscillations are produced in the computational domain.
(a)
(b)
5. Conclusion
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.
Data Availability
The authors confirm that the data supporting the findings of this study are available within the article (No. 7343716).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
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).