An analytic solution of Burgers equations (1) was given by Fletcher  using the Hopf-Cole transformation as follows:
From the analytic solution (2), we obtain . If , we have and . Because the analysis process is very long, we divide it into two parts for the readers’ convenience.
First, considering the first equation of (2), we have
Now for the second equation of (2), we can also get
We will do some appropriate transformation to (3) and (4) in the following analysis.
2.1 Explicit exact-difference scheme
In this section, we will give the explicit finite-difference scheme for (1).
2.1.1 Discrete scheme for the first equation
Using (3a) minus (3), we obtain
By defining , using (5), we get the following result:
If we set and use (6), we obtain
And if we use (3) minus equation (3c), we get
Setting , we immediately obtain
According to (9), if we define , we have
Based on (7) and (10), we have the following result:
Because , , , we have
Equation (12) can be seen as a discrete format of the first equation of (1). This discrete format is an explicit scheme. In the following section, we will consider the discrete scheme for the second equation of (1).
2.1.2 Discrete scheme for the second equation
Using (4a) minus (4), we obtain
Setting , then according to (13) we have the following equation:
In the same way as , we define , use (14), and we have
Equation (4) minus (4c) yields
Setting , we obtain
Define , on the basis of (17), we obtain the following result:
On the basis of (15) and (18), we have
For , so , set , we have
Equation (20) is the explicit discrete format of the second equation in (1).
2.2 Implicit exact-difference scheme
We will give the implicit format of (1) in the following analysis.
2.2.1 Discrete scheme for the first equation
Furthermore, if we use (3) minus (3b)
According to (21), we assume . We obtain another result:
If we substitute (22) into , we will get the following result:
Equation (3d) minus (3) yields
By setting , we have
Referring to (25), we get the following equation:
And on the basis of (23) and (26), we have
Assume . We already know , so , and we obtain
Equation (28) is the implicit discrete format for the first equation of (1). Now, we will give the implicit scheme for the second equation of (1).
2.2.2 Discrete scheme for the second equation
If we use (4) minus (4b), we immediately get
Set , then we have
Using (30), we obtain
Using (4d) minus (4), we immediately get
Setting , we have
According to (33), we have
According to (31) and (34), we obtain
Because , so , setting , we have
Now, we can find that (36) is the implicit discrete format of the second equation in (1). In the next step, we will give the exact finite-difference schemes.
2.3 The exact finite schemes for coupled Burgers equation
We denote the discrete approximation of and at the mesh point by and , respectively, ; ; , where is the mesh size in x direction and is the mesh size in y direction, and τ represents the increment in time. Set , we write as
For , , , , based on (12) and (20), we have the explicit difference scheme
For , , , , according to (28) and (36), we can obtain an implicit difference scheme
Theorem 2.1 The difference schemes (38) and (39) are the exact explicit and implicit difference schemes for (1), where , , are the expressions in Sections 2.1 and 2.2.
Proof We can see the analysis from Section 2.1 and Section 2.2 that the two schemes are explicit and implicit exact finite schemes. □