MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/73437167343716Research ArticleNonlinear Finite Volume Scheme Preserving Positivity for 2D Convection-Diffusion Equations on Polygonal Mesheshttps://orcid.org/0000-0002-5290-0831LanBin12DongJianqiang3YangZaoli1School of Mathematics and Information ScienceNorth Minzu UniversityYinchuan 750021Chinanun.edu.cn2The Key Laboratory of Intelligent Information and Big Data Processing of Ningxia ProvinceNorth Minzu UniversityYinchuan 750021Chinanun.edu.cn3College of Civil EngineeringHefei University of TechnologyHefei 230009Chinahfut.edu.cn20202182020202012052020200720202907202021820202020Copyright © 2020 Bin Lan and Jianqiang Dong.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Natural Science Foundation of Ningxia Province2020AAC032352020AAC03233General School Level Project12020000153National Natural Science Foundation of China1160101311772165First-Class Disciplines Foundation of NingxiaNXYLXK2017B09
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 . Pert  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  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 . Some nonlinear methods have been developed  for general diffusion or convection-diffusion equations, which guarantee the positivity on general or distorted meshes for general tensor coefficients.

Bertolazzi and Manzini  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.  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. , 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  and convective flux , 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 u=ux:(1)κuvu=f,inΩ,(2)u=g,onΩ,where Ω is a bounded polygonal domain in × with boundary ∂Ω, κ=κx is a known diffusive coefficient, and v=vx is a velocity vector field.

Assume that the functions vx,fx, and gx satisfy the constraints listed as follows:(3)v0,vC1Ω¯2,fL2Ω,gH1/2ΩCΩ,and there are two positive constants λ1 and λ2 such that(4)λ1ξ2κxξξλ2ξ2,ξ×.

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 .

We use a mesh on Ω made up of arbitrary convex polygon cells. The 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=KLΕ. The cell-edge σ is also denoted by AB, and the midpoint of σ is denoted by M. Moreover, we denote P1 and P2 are two adjacent midpoints of cell K (Figure 1).

Local stencil 1.

Let nK,σ (or nL,σ) be the unit outer normal vector on the cell-edge σ of cell K (or L), κT be the transpose of matrix κ, and K be the set of all edges of cell K. Denote εint=εΩ and εext=εΩ. Denote h=supKTmK1/2, where mK is the area of cell K.

Integrating (1) over the cell K, we obtain(5)σΕKK,σ+GK,σ=Kfxdx,where the diffusive and convective flux are defined as(6)K,σ=σuxκTxnK,σdl,(7)GK,σ=σvuxnK,σdl.

3. Construction of the Scheme3.1. The Diffusive Flux

Following the idea in the study by Sheng and Yuan , 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 uM. First, we give a brief review of the construction.

Figure 1 shows that a ray originating at the point K along the direction κTnK,σ must intersect one segment connecting two neighboring midpoints of edge of cell K, where the two midpoints are denoted by P1 and P2, and the cross point is denoted by O1. Similarly, a ray originating at the midpoint M along the direction κTnK,σ must intersect one certain KP4, where P4 must be one vertex of σ, and the cross point is denoted by O2.

Let tKP1,tKP2,tMK, and tMP4 be some unit tangential vectors along their corresponding directions, respectively. θii=1,,4 are some corresponding angles. Hence, we established the following relations:(8)κTnK,σκTnK,σ=sinθ2sinθ1+θ2tKP1+sinθ1sinθ1+θ2tKP2,(9)κTnK,σκTnK,σ=sinθ4sinθ3+θ4tMK+sinθ3sinθ4+θ4tMP4.

Substituting equation (8) into equation (6) and neglecting the high-order terms, we have(10)F¯1=κKTnK,σσsinθ2sinθ1+θ2uP1uKKP1+sinθ1sinθ1+θ2uP2uKKP2=a1uKuP1+a2uKuP2,where(11)a1=κKTnK,σσKP1sinθ2sinθ1+θ2,a2=κKTnK,σσKP1sinθ1sinθ1+θ2.

Similarly, substituting equation (9) into equation (6), we have(12)F¯2=κKTnK,σσsinθ4sinθ3+θ4uKuMMK+sinθ3sinθ3+θ4uP4uMMP4=a3uKuM+a4uP4uM,where(13)a3=κKTnK,σσMKsinθ4sinθ3+θ4,a4=κKTnK,σσMP4sinθ3sinθ3+θ4.

Combined with equations (10)–(12), the discrete normal flux on σ can be defined as follows:(14)FK,σ=μ1F¯1+μ2F¯2=μ1a1uKuP1+μ1a2uKuP2+μ2a3uKuM+μ2a4uP4uM,where μ1 and μ2 are some nonlinear coefficients with μ1+μ2=1, which will be given later.

In order to assure μ1 and μ2 are positive, two additional parameters ω1 and ω2 are introduced later. According to the different positions of P1 and P2, three cases exist.

We assume that uP1uM and uP2uM, the normal flux (14) can be rewritten as(15)FK,σ=μ1a1+a2+μ2a3uKμ2a3+a4uMμ1a1uP1+a2uP2+μ2a4uP4=μ1a1+a21+ω1+μ2a3uKμ2a3+a41+ω2uMμ1a1uP1+ω1uK+a2uP2+ω1uK+μ2a4uP4+ω2uM.

In order to obtain the two-point flux approximation, the last two terms of the above expression should vanish; hence, μ1 and μ2 are given as follows:(16)μ1=a4uP4+ω2uMa1uP1+ω1uK+a2uP2+ω1uK+a4uP4+ω2uM,μ2=a1uP1+ω1uK+a2uP2+ω1uKa1uP1+ω1uK+a2uP2+ω1uK+a4uP4+ω2uM.

Hence, (15) can be expressed as follows:(17)FK,σ=AK,σ,1uKAK,σ,2uM,where(18)AK,σ,1=μ1a1+a21+ω1+μ2a3,AK,σ,2=μ2a3+a41+ω2.

In order to assure μ1>0 and μ2>0, two parameters ω1 and ω2 can be chosen such that(19)a1uP1+ω1uK+a2uP2+ω1uK0,a4uP4+ω2uM0.

If(20)a1uP1+ω1uK+a2uP2+ω1uK=a4uP4+ω2uM=0.

We let μ1=μ2=1/2.

If uP1=uM or uP2=uM, equation (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(21)FL,σ=AL,σ,1uLAL,σ,2uM.

Using the continuity of normal flux FK,σ+FL,σ=0 on edge σ, we obtain(22)uM=AK,σ,1uK+AL,σ,1uLAK,σ,2+AL,σ,2.

Substitute equation (22) into equation (17) to obtain the nonlinear two-point diffusive flux on σ=KL:(23)FK,σ=AK,σuKAL,σuL,where AK,σ=AK,σ,1AL,σ,2/AK,σ,2+AL,σ,2, and AL,σ=AK,σ,2AL,σ,1/AK,σ,2+AL,σ,2.

From the computation of vertex unknowns, a method with second-order accuracy has been proposed in the study by Sheng and Yuan . We know that uM>0 as long as uK>0 and uL>0 in equation (22).

3.2. The Convective Flux

We focus on the expression of convective flux in equation (7) for σ=KLεint and obtain(24)GK,σ=uMσvnK,σdl+Oh2=uMvK,σ+vK,σ+Oh2,where uM is the value of midpoint M on the cell-edge σ, and(25)vK,σ+=12vK,σ+vK,σ,vK,σ=12vK,σvK,σ,vK,σ=σvnK,σdl.

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 vK,σ+ and negative part vK,σ. Moreover, the property of upwind is also considered.

Neglecting the high-order terms, we have the following approximate expression of the upwind formula :(26)GK,σuMvK,σ+vK,σ.

In order to approximate the continuous flux GK,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 M1,M2 are the other two midpoints adjacent to it. We denote SMM1K be the area of triangle MM1K and define(27)u¯K=Mσ,σKωMuM,where ωM=SMM1K+SKM2M1/Mσ,σεKSMM1K+SKM2M1. For a special case (Figure 3), we set the cell K as a triangle and define u¯K=ωMuM+ωM1uM1+ωM2uM2, where(28)ωM=s1s,ωM1=s2s,ωM2=s3s,s=s1+s2+s3,s1=SMM1K+SKM2M1,s2=SMM1K+SKM1M21,s3=SKM1M2+SMKM21.

Local stencil 2.

Local stencil of triangle.

It is obvious that u¯K is a second-order approximation to uK, i.e., uKMσ,σKωMuM=Oh2 if the solution uC2K.

Then, the approximation of GK,σ on cell-edge σ can be defined as follows.

For σint, we define(29)GK,σ=BK,σuKBL,σuL,where(30)BK,σ=vK,σ+uMu¯K0,BL,σ=vL,σ+uMu¯L0.

For σext, we define(31)GK,σ=BK,σuKbK,σ,where(32)BK,σ=vK,σ+uMu¯K0,bK,σ=vK,σgM.

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:(33)σεKFK,σ+GK,σ=KFK+σεKεextAK,σ,2uM+bK,σ,KJ.(34)uMi=gMi,MiΩ,where fK=fK and gMi=gxMi.

Let U be the discrete unknown vector and AU be the coefficient matrix. A nonlinear algebraic system of the schemes (33) and (34) can be formed: AUU=F. The AU is assembled by the coefficients of diffusive term FK,σ and convective term GK,σ. We use the Picard nonlinear iteration method to solve the system: choose a small value non>0 and initial vector U0>0 and repeat for nonlinear iteration index s=1,2,,:

Solve AUs1Us=F

Stop if AUsUsF2ΕnonAU0U0F2

The linear algebraic system with coefficient matrix AUs1 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 εlin.

3.4. The Algorithm

In this subsection, we describe the detailed algorithm.

Step 1. Initialize U0>0, εnon, and εlin.

Step 2. When s=0,

compute AK,σ0 and BK,σ0;

compute initial residual AU0U0F2.

Step 3. When s=1,2,,

solve AUs1Us=F;

correct Us, see Remark 1;

compute uPs and correct uMs, Mσ,σΕ,PN, see Remark 2;

compute AK,σs and BK,σs;

compute residual AUsUsF2;

whether AUsUsF2εnonAU0U0F2, if true, then go to (Step 4), otherwise, go to (Step 3).

Step 4. Stop.

Remark 1.

For uKs, if uKs>maxuMs1,Mσ,σΕK, let uKs=u¯Ks1.

Remark 2.

The value of uMs can be obtained by equation (22). If uMs>maxuKs,uLs,uAs,uBs, where the common edge σ of two cells K and L is also denoted by AB (Figure 1). Let uMs=uMs1.

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 F0,U00, and linear systems in Picard iterations are solved exactly. Then,(35)Us0,s=1,2,3,,.

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 umax=max0,uK,uM,KT,Mε.

We assume uK0=umax. Using Remark 1, we know that uK0maxuM,Mσ,σεK0.

4. Numerical Experiments

In order to demonstrate the accuracy and robustness of the scheme, we test several problems and take εnon=1.0e6 and εlin=1.0e10. The convergence order can be obtained by the following formula:(36)Order=logErrorN1/ErrorN2logN2/N1,where N1 and N2 represent different number of cells, and ErrorN1 and ErrorN2 are the corresponding L2 errors.

4.1. The Problem with Anisotropic Diffusion Tensor

Consider the problems (1) and (2) with Dirichlet boundary condition on Ω=0,1×0,1, and take v=1,1T. The exact solution is(37)ux,y=cosπx2ey,and the diffusion coefficient is(38)κ=100ε00ε.

First, we test the accuracy of our scheme on random quadrilateral meshes shown in Figure 4. Table 1 gives the corresponding L2 error and the numbers of nonlinear iteration numbers itnon# 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. The average number of nonlinear iterations is 36.8 when ε=106. However, for ε=1.0, the corresponding number increases while the number of cell increases.

Accuracy for the problem with anisotropic coefficient.

Cells1445762304921636864
ε=1.0ε2u3.11E37.50E42.09E45.46E51.42E5
Order2.051.841.941.94
ε2F1.06E+03.94E11.76E18.28E23.96E2
Order2.071.542.030.99
itnon#5585116126153

ε=106ε2u5.28E31.24E33.18E48.06E52.13E5
Order2.091.961.981.92
ε2F7.93E31.99E35.31E41.34E43.57E5
Order2.071.542.030.99
itnon#3640363339
4.2. The Problem with Discontinuous Coefficient

Consider the problems (1) and (2) with Dirichlet boundary condition on Ω=0,1×0,1, and take v=1,1T. The exact solution is(39)ux,y=sinπ2x+sinπ2y,x<12,2cπx1/24+sinπ2y+22,x12,and the diffusion coefficient is(40)κ=c0×ε,x<12,ε,x12,where c0=40.

We test this problem on random triangle meshes shown in Figure 5. The numerical results with a different parameter ε=1.0,106 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 ε=1.0 and ε=1.0,106, respectively.

The random triangle meshes.

Accuracy for the problem with discontinuous coefficient.

εCells288115246081843273728
1.0ε2u4.87E41.39E43.82E58.68E62.13E6
Order1.811.862.142.03
ε2F7.83E23.04E21.39E25.51E32.50E3
Order2.071.542.030.99
itnon#2324262626

106ε2u4.54E41.18E43.13E57.93E62.01E6
Order1.941.911.981.98
ε2F6.60E41.63E44.15E51.04E52.63E6
Order2.071.542.030.99
itnon#2324232323

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.  and more than 2.0 in the study by Lan et al.  and our new scheme. However, the computed results in our scheme can approximate the exact solution more well.

Comparison of accuracy with discontinuous coefficient ε=105.

MethodMeshCells2561024409616384
Zhang et al. Uniformε2u1.10E − 13.98E-21.42E − 25.02E − 3
Order1.471.491.50
Randomε2u1.10E − 13.95E-21.38E − 25.11E − 3
Order1.481.521.43

Lan et al. Uniformε2u4.23E − 39.77E-42.31E − 45.50E − 5
Order2.122.082.07
Randomε2u6.87E − 31.57E-33.78E − 48.96E − 5
Order2.132.052.08

PresentUniformε2u9.34E − 41.82E − 43.63E − 58.43E − 6
Order2.362.322.11
Randomε2u9.16E − 41.89E − 44.04E − 59.85E − 6
Order2.282.232.04
4.3. Positivity of Numerical Solutions

Now, we consider the problem (1)-(2) in the unit square Ω=0,12 with homogeneous Dirichlet boundary conditions. Take v=1,1T, and set(41)κ=cosθsinθsinθcosθk100k2cosθsinθsinθcosθ,f=1,ifx,y38,582,0,otherwise.

We take k1=1,k2=100, and θ=5π/6.

The analytical solution ux,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. The corresponding distribution is similarly shown in Figure 6), and the numerical solution is given in Figure 7. The minimum value is umin=0 and the maximum value is umax=6.7603×104, 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 umin=0 and umax=6.7582×104.

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.

4.4. Nonphysical Oscillations

We also consider the last nonsmooth anisotropic solution and compute it on the random quadrilateral meshes. Here, we reset f=0. The computational domain is a unit square with a hole, Ω=0,124/9,5/92, so that the boundary Ω is composed of two disjoint parts Γ0 and Γ1 as shown in Figure 8 where the number of cell is 72×72. Γ0 is the exterior boundary, and Γ1 is the interior boundary. We set g=0 on Γ0 and g=2 on Γ1.

The random quadrilateral meshes with a hole.

The numerical solutions on the random quadrilateral meshes are shown in Figure 9. The computed results show that umin=6.94E13 and umax=1.98. It means that the minimum 0 is attained on the Γ0, and the maximum 2 is attained on the Γ1. So, these computed results illustrate that our scheme can avoid the numerical solution beyond the upper bound and does not produce any nonphysical oscillations.

The numerical results on the random quadrilateral meshes.

Then, the computed results without the correct method in Remarks 1 and 2 are shown in Figure 10, and umin=3.87E6 and umax=1.91. However, the numerical oscillations are produced in the computational domain.

The numerical results without corrections on the random quadrilateral meshes.

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).

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