In this part we will consider the following one dimensional convection-diffusion equation,

{ − ε u x x + α u x = f ( x ) ,   x ∈ Ω , u | ∂ Ω = g . (11)

where the diffusion coefficient ε and convection coefficient α are constants. f ( x ) is the cource term.

Using the central finite difference method, the second order derivative term u x x has the following approximate format

u x x | x i = u i + 1 − 2 u i + u i − 1 h 2 + O ( h 2 ) . (12)

For the first order derivative term u x , there are many different ways to approximate it, such as the central finite difference method, the upwind finite difference method, and the modified finite difference method, etc. Finally, we will discuss several different numerical algorithms and compare the advantages and disadvantages of these algorithms.

Example 1.

{ − ε u x x + u x = 1 , u ( 0 ) = 0 , u ( 1 ) = 0. (13)

In this case, we will give the corresponding numerical solution by finite difference method. In the interval [ 0,1 ] we will use n + 1 nodes and give a equidistant discretization, i.e. each subinterval the length h = 1 n . Then each point of the interval can be expressed as x i = i n , i = 1 , 2 , ⋯ , n .

For the first order derivative term u x , we use

u x | x i = u i + 1 − u i − 1 2 h + O ( h 2 ) . (14)

Then for this problem, the following central finite difference scheme is obtained.

− ε u i + 1 − 2 u i + u i − 1 h 2 + u i + 1 − u i − 1 2 h = 1 , (15)

The trunction error is O ( h 2 ) .

Let’s denote 2 ε h 2 = α , − ε h 2 + 1 2 h = β , − ε h 2 − 1 2 h = γ , then we can write the discrete form of convection-diffusion equation as the following expression,

( α β 0 ⋯ 0 0 0 γ α β ⋯ 0 0 0 0 γ α ⋯ 0 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 0 0 0 ⋯ γ α β 0 0 0 ⋯ 0 γ α ) ( u 1 u 2 u 3 ⋮ u n − 2 u n − 1 ) = ( 1 − γ u 0 1 1 ⋮ 1 1 − β u n ) . (16)

Figures 2-5 show the numerical and exact solutions of the convection diffusion equation for different ε respectively. Figure 2 and Figure 3 show that when ε > 1 , the numerical solution agrees very well with the exact solution, and

there are no significant errors from the plots. Figure 4 and Figure 5 show that the numerical and exact solutions agree well with relatively small errors in the left half of the interval, but the errors become larger in the right side of the interval, especially near the right endpoint of the interval [ 0,1 ] .

As shown in Table 1, the absolute error between the exact and numerical solutions when ε takes different values, i.e. ε = 100 , 2 , 1 20 , 1 100 . When the coefficient ε is less than 1, the numerical solution of the convective diffusion equation deviates significantly from the exact solution at the discrete points on the right side of the interval, and the numerical format has a significant oscillation at these points. When ε is less than 1, the smaller ε is, the larger the error is.

The traditional differential format produces numerical oscillations when the coefficient is less than one, especially when the coefficient is much less than one. This is mainly because, when the coefficient is much less than 1, the equation at this point becomes a convective dominance problem.

In order to overcome numerical oscillations, scholars have proposed upwind finite difference method (16), corrected finite difference method (17), characteristic finite difference method and other methods based on the traditional finite difference method.

Upwind finite difference method,

− ε u i + 1 − 2 u i + u i − 1 h 2 + u i − u i − 1 h = 1. (17)

Corrected finite difference method,

− ε u i + 1 − 2 u i + u i − 1 h 2 + [ λ u i − u i − 1 h + ( 1 − λ ) u i + 1 − u i − 1 2 h ] = 1. (18)

For the corrected finite difference method, is a combination of central difference method and upwind difference method with weight λ and 1 − λ separately, where 0 ≤ λ ≤ 1 .

Similarly, the traditional finite element method also has numerical oscillations for the convection dominant equation. Many scholars have proposed a series of correction techniques to combine the finite element method with other methods, and proposed the characteristic finite element method, characteristic hybrid element method, etc.

Figure 6 and Figure 7 show the numerical solutions of various different algorithms when ε takes different parameters. For the central differential format, the numerical oscillation is obvious when ε is smaller. The upwind differential format and the modified upwind differential format can overcome the numerical oscillation phenomenon, but the accuracy of these methods is not good enough. Therefore, an algorithm that can avoid numerical oscillations and at the same time obtain high numerical accuracy is necessary for solving the convection-diffusion equation.

Table 2 shows the absolute errors between the numerical solution and the exact solution obtained with different numerical algorithms. Error₁, Error₂, and

Error₃ in the table are the absolute errors between the analytical solution and the numerical solution obtained in the central difference format, upwind difference format, and the corrected upwind difference format, respectively. For the same points as shown in Table 2, especially at the points near the right end of the interval, Errors₂ and Error₃ are significantly smaller than Error₁, which also means that there is no numerical oscillation for the corresponding numerical format.

Algorithm 1: Chebyshev spectral method for one dimentional convection-diffusion equation

For the convection-diffusion equation, we can obtain the following numerical format, using the Chebyshev spectral method,

− ε D 2 u + D u = f , (19)

where D is the Chebyshev matrix, f = ( 1,1, ⋯ ,1 ) T . If we denote L as the coefficient matrix − ε D 2 + D , then we have

L u = f . (20)

Next, we consider the boundary conditions u ( 0 ) = 1 , u ( 1 ) = 0 . We need to make a small modification to the coefficient L. After replacing the first and last row of the matrix L with ( 1,0, ⋯ ,0,0 ) , ( 0,0, ⋯ ,0,1 ) respectively, we denote the new coefficient matrix as L ^ . For the right-hand side term f of the Equation (18), we also need to modify the first and last terms to 0, we denote the modified f as f ^ . Finally, we get the numerical format for the convection-diffusion equation with Chebyshev spectral method,

L ^ u = f ^ . (21)

Figure 8 shows the numerical solutions given by the Chebyshev spectral method and several of the previously discussed finite difference methods. It is clear that the Chebyshev spectral method does not have numerical oscillations while being able to obtain higher accuracy. This is mainly because the Chebyshev points are not uniformly distributed and are more concentrated at the two ends of the interval, so there is no numerical oscillation near the interval endpoints.

In Table 3, Error₁, Error₂, Error₃ and Error_ch are the errors of cerntral difference method, upwind difference method, corrected difference method and Chebyshev spectral method respectively and 21 nodes are used for each format. From Table 3, it is clear that the numerical solution calculated with the central difference format has the largest error and the numerical solution calculated by the Chebyshev spectral method has the smallest error.

Table 4 shows the absolute errors between the numerical results calculated by the Chebyshev spectral method and exact solutions when different parameters are taken. It is obvious that when the coefficients are fixed, the larger the number of nodes, the smaller the error, meanwhile, when the coefficients are very small there is no numerical oscillation. When the node N is large enough, the absolute error values in the table are already very small and exceed the minimum value of the system, so some of the data values in the table are 0.

In this part we will focus on the two dimensional convection-diffusion equation. The specific equation format is as follows,

− ε ( u x x + u y y ) + α ( x , y ) u x + β ( x , y ) u y + u = f ( x , y ) ,   ( x , y ) ∈ Ω , (22)

where the diffusion coefficient ε is a constant, α ( x , y ) , β ( x , y ) are the convection coefficients, and f ( x , y ) is the source term. The convection-diffusion satisfies Dirichlet boundary condition, i.e. u | ∂ Ω = g ( x , y ) .

Both in x and y directions we use the central difference method respectively, we have

u x x | x i , y j = u i + 1 , j − 2 u i , j + u i − 1 , j h 2 + O ( h 2 ) , (23)

u y y | x i , y j = u i , j + 1 − 2 u i , j + u i , j − 1 h 2 + O ( h 2 ) , (24)

u x | x i , y j = u i + 1 , j − u i − 1 , j 2 h + O ( h 2 ) , (25)

u y | x i , y j = u i , j + 1 − u i , j − 1 2 h + O ( h 2 ) . (26)

Then we have the following numerical scheme.

  − ε ( u i + 1 , j − 2 u i , j + u i − 1 , j h 2 + u i , j + 1 − 2 u i , j + u i , j − 1 h 2 )   + α i , j u i + 1 , j − u i − 1 , j 2 h + β i , j u i , j + 1 − u i , j − 1 2 h + u i , j = f i , j . (27)

In the above numerical format α i , j = α ( x i , y j ) , β i , j = β ( x i , y j ) , f i , j = f ( x i , y j ) . Both in x and y directions, we use the same step size h.

Example 2.

{ − ε u x x + u y y + 1 1 + y u x = f ( x , y ) ,   ( x , y ) ∈ [ 0 , 1 ] × [ 0 , 1 ] , u ( x , 0 ) = e − x + 2 − 1 ε , u ( x , 1 ) = e 1 − x + 2 , u ( 0 , y ) = e y + 2 − 1 ε ( 1 + y ) 1 + 1 ε , u ( 1 , y ) = e y − 1 + 2 − 1 ε ( 1 + y ) 1 + 1 ε . (28)

On the right hand side of the above equation f ( x , y ) = ( − 2 ε − 1 1 + y ) e y − x − 2 − 1 ε ( 1 + 1 ε ) ( 1 + y ) 1 ε − 1 .

The numerical solution of Example 2 using the central finite difference algorithm (26) is shown in Figure 9, and the analytic solution of the corresponding problem is shown in Figure 10. It is clearly that when ε is small, there is a significant difference between the numerical solution and the exact solution at the points near the end of the definition domain. Figure 12 shows the absolute errors of the numerical and exact solutions at the nodes, which are smaller when x and y are near the left endpoint 0 and have become quite large when x and y are near the right endpoint 1.

Figure 11 and Figure 12 show the errors between the numerical solution given by the central difference method and the exact solution when the diffusion coefficient ε takes different values. When the diffusion coefficient ε is large, the numerical solution given by the central difference algorithm agrees better with the exact solution, but when the diffusion coefficient is small, the error between the numerical solution and the exact solution is larger.

Table 5 shows the maximum error in the definition domain when the nodes and diffusion coefficients take different values. From Table 5 we can clearly see that when the diffusion coefficient ε ≥ 1 , the maximum error is relatively small regardless of the number of nodes, but when the diffusion coefficient ε is small, as shown in the table when ε = 1 100 or ε = 1 200 , the maximum error has reached an unacceptable level even if the number of nodes is large.

Algorithm 2: Chebyshev spectral method for two dimenditional convection- diffusion equation.

For the Chebyshev derivative matrix D, as discussed in the previous section, we will not expand the discussion in detail here. If we arrange the solution u into a column vector along the y direction. Then u x x + u y y can be expressed as ( D 2 ⊗ I + I ⊗ D 2 ) u , where ⊗ is the Kronecker product. The first order term 1 1 + y u x can be writen as d i a g ( 1 1 + y 0 , ⋯ , 1 1 + y n ) ( D ⊗ I ) U . Similarly, for the right end term f of the equation we rearrange it and denote it as F. Then for two dimenditional convection-diffusion Equation (27) we have the following Chebyshev spectral numerical format.

( − ε ( D 2 ⊗ I + I ⊗ D 2 ) + d i a g ( 1 1 + y 0 , ⋯ , 1 1 + y n ) ( D ⊗ I ) ) U = F . (29)

If we denote − ε ( D 2 ⊗ I + I ⊗ D 2 ) + d i a g ( 1 1 + y 0 , ⋯ , 1 1 + y n ) ( D ⊗ I ) as L, then the above format can be simply written as

L U = F . (30)

However, this Format (28) or (29) only considers the internal nodes, and the boundary conditions are not yet considered. If we take the boundary conditions into account, then we need to make a small modification to (28) or (29), i.e., replace the coefficient matrix and the corresponding row of the right terminal term with the corresponding relationship at the boundary. If we denote the modified coefficient matrix and the right terminal term as L ^ and F ^ , respectively, then we have,

L ^ U = F ^ . (31)

We use the Chebyshev spectral method given above to calculate the numerical solution of Example 2. Figure 13 and Figure 14 show the errors when ε is taken as 10 and 1 100 respectively, and 21 nodes are taken in the x and y directions.

Figure 13 shows that when ε = 10 , the maximum absolute error is less than 1.5 × 10⁻⁹. When the diffusion coefficient is ε = 1 100 , the absolute error of the numerical solution given by the spectral method becomes larger but is still less than 4 × 10⁴. This result is much better than the numerical solution given by the central difference format, and there is no numerical oscillation.

Table 6 gives the absolute errors of the numerical solutions obtained by the Chebyshev spectral method for example 2 when different parameters are taken.

The errors are all better than the finite difference method when the diffusion coefficient is smaller, which also shows that the spectral method is better adapted than the finite difference method for solving the convective dominance equation.