The numerical solutions for the nonhomogeneous Burgers' equation with the generalized Hopf-Cole transformation

1.   Introduction

Burgers' equation plays an important role in analyzing fluid turbulence since it has much in common with the Navier-Stokes equation. It was introduced by an English mathematician H. Bateman in 1915 ^[1] with its corresponding homogeneous boundary conditions as

where $\mu > 0$ is the kinematic viscosity. A Dutch physicist J.M. Burgers explained the mathematical simulation of turbulence with the help of Eqs (1.1a)–(1.1c) in 1948 ^[2], which made this equation famous. In honor of his work, the equation is named omit the Burgers' equation. In order to efficiently solve Eqs (1.1a)–(1.1c), E. Hopf ^[3] and J.D. Cole ^[4] independently introduced a transformation to convert Eqs (1.1a)–(1.1c) into a heat equation with Neumann boundary condition, which made the exact solution explicitly for arbitrary initial conditions. The transformation was well-known as the Hopf-Cole transformation $u(x, t) = -2\mu \frac{w_x}{w}$, where $w(x, t)$ satisfied the following equation

During the last few decades, many significant efforts have been carried out oriented towards the robust numerical schemes for Burgers' equation, which forms a benchmark problem in parallel and distributed computation for the partial differential equations solvers ^[5,6]. Among the various solvers, there are analytical methods involving classical Hopf-Cole transformation ^[3,4]. There are also many numerical solvers involving finite difference methods ^{[7,8,9,10,11]}, finite element methods ^[12,13], spectral methods ^[14,15,16] and classification to name a few. A nice and systematic literature for Burgers' equation is referred to in the recent review in ^[17].

In the paper, based on the Hopf-Cole transformation, we devote to designing an effective numerical method for the Burgers' equation with the nonhomogeneous source function and nonhomogeneous boundary conditions ^[18] as

where $\varphi(x)$, $\alpha(t)$ and $\beta(t)$ are arbitrary given smooth functions. $f(x, t)$ is the trivial source function. In order to satisfy the compatibility condition, we require $\varphi(0) = \alpha(0)$ and $\varphi(L) = \beta(0)$.

The fundamental difficulty lies in analyzing the prior estimate for solving problem Eqs (1.3a)–(1.3c) since the analysis of problems with the derivative boundary conditions are totally different from that with the Robin boundary conditions. In this paper, the emerging difficulties are overcome via the help of the reduction order method and the principle of boundary homogeneity for the transformed problem.

The main novelty of this paper aims at we convert Eqs (1.3a)–(1.3c) into an equivalent heat equation with the derivative boundary conditions for the first time, in which Neumann boundary conditions and Robin boundary conditions can be viewed as special cases. The concrete contributions are listed as follows

● The generalized exponential transformation links the classic Hopf-Cole transformation to the exponential transformation for the constant convection term or variable convection term. In fact, the classic Hopf-Cole transformation can be viewed as the special case of the generalized exponential transformation, see the formula Eq (2.1) in Section 2. When the first $u$ in nonlinear convection term $uu_x$ is considered as a constant coefficient, the application of the generalized exponential transformation is referred to ^[19,20]. And when it is viewed as a variable coefficient, its application is referred to ^[21,22].

● We strictly show that the Burgers' equation with nonhomogeneous boundary conditions and nonhomogeneous right-hand side term is equivalent to a heat equation with the derivative boundary conditions.

● An equivalent box scheme is offered for easy implementation. The solvability and convergence of the established box scheme are analyzed in detail by the technical energy argument.

● Numerical errors and the convergence orders for the homogeneous and nonhomogeneous problems are displayed and verify the effectiveness of the proposed box scheme.

● Compared with other papers on the Burgers equation ^[23,24], the main difference or advantage of the method presented in this paper is that we deal with the non-homogeneous boundary problem, while most other methods are only suitable for solving burgers equation with homogeneous boundary.

The rest of the paper is arranged as follows. Section 2 presents the equivalent form for the nonhomogeneous Burgers' equation based on the generalized Hopf-Cole transformation. Section 3 is the main part of the paper, which focuses on the analysis and derivation of the difference scheme. More concretely, it involves some useful notations and lemmas, the reduced order method, a priori estimate of the difference scheme, solvability and convergence. The numerical experiments are carried out in Section 4, followed by some conclusions in Section 5.

To facilitate numerical analysis in what follows, we suppose there exists a constant $c_0$ such that

throughout the whole paper, .

2.   An equivalent form

Introducing the generalized exponential transformation

and taking the derivative of both sides with respect to $x$ in Eq (2.1), then noting the boundary conditions Eq (1.1c), we have the classical Hopf-Cole transformation ^[3]

Via the help of Eq (2.2), we have

Substituting Eq (2.2) and Eq (2.3) into Eq (1.3a), we have

or

Furthermore, we have

Integrating for $x$ from 0 to $x$ on both sides of Eq (2.4), we have

where

Multiplying Eq (2.5) by $\exp(-\int_0^t q(s)ds)$ on both sides, we have

Let

Then, we have

In other words, for the arbitrary $q(t)$, $u(x, t)$ is independent of $q(t)$. Thus, we take $q(t) = 0$ for brevity. Meanwhile, Eq (2.5) is simplified as

By Eq (2.1), we obtain the initial condition

Noticing Eq (2.6), we have

Thus, the left boundary condition reads

Similarly, we have the right boundary condition

Above procedures are invertible, thus Eqs (1.3a)–(1.3c) is equivalent to Eqs (2.7)–(2.10).

Remark 1. We make some explanations about the exponential transformation Eq (2.1) and the boundary conditions Eqs (2.9)–(2.10).

$(a)$ The classical Hopf-Cole transformation Eq (2.2) can be viewed as a special case of Eq (2.1). The first $u(x, t)$ of $uu_x$ in Eq (1.3a) is supposed to be a "ghost constant coefficient". In this hypothetical situation, the constant coefficient convection term and variable coefficient convection term (e.g., ^[19,22]) are diminished by the generalized exponential transformation Eq (2.1).

$(b)$ The boundary conditions Eq (2.9) and Eq (2.10) are called derivative boundary conditions. To assure that solution of Eqs (2.7)–(2.10) is stable and unique, one usually requires $\alpha(t) \leqslant 0, \; \beta(t) \geqslant 0$. Under the above constraint, Eq (2.9) and Eq (2.10) are called Neumann boundary conditions if $\alpha(t)+\beta(t)\equiv0$ and Robin boundary conditions if $\alpha(t)+\beta(t) \not\equiv 0$, see e.g., ^[25]. The arbitrariness of $\alpha(t)$ and $\beta(t)$ makes the numerical analysis of the problem with derivative boundary conditins difficult compared with that with Robin boundary conditions. In present paper, we will analyze the general cases.

$(c)$ The right-hand side function $F(x, t)$ in Eq (2.7) can be computed by Simpson formula numerically when it can not be expressed explicitly by the elementary functions.

3.   The derivation and analysis of the difference scheme

3.1.   Notations and lemmas

Before introducing the finite difference scheme, we divide the domain $[0, L]\times[0, T]$. Take positive integers $M$ and $N$ and let $h = L/M$, $\tau = T/N$. Denote $x_i = ih$, $0 \leqslant i \leqslant M$; $t_k = k\tau$, $0 \leqslant k \leqslant N$; $\Omega_{h} = \{x_i\, |\, 0 \leqslant i \leqslant M\}$, $\Omega_{\tau} = \{t_k\, |\, 0 \leqslant k \leqslant N\}$, $\Omega_{h\tau} = \Omega_h \times \Omega_{\tau}$. For any grid function $v = \{u_i^k\, |\, 0 \leqslant i \leqslant M, 0 \leqslant k \leqslant N\}$ defined on $\Omega_{h\tau}$, we denote

The following two lemmas come from ^[25].

Lemma 1. Let $\{F^k\}_{i = 1}^{\infty}$ and $\{g^k\}_{i = 1}^{\infty}$ be two non-negative sequences and satisfy

then

Lemma 2. Let $u = (u_0, u_1, \cdots, u_M)$ be a discrete function on $\Omega_h$, then for any $\varepsilon > 0$, we have

3.2.   The reduced order method for the difference scheme

In the previous section, we have converted Eqs (1.3a)–(1.3c) into an equivalent heat conduction equation with the derivative boundary conditions by the generalized Hopf-Cole transformation. Then, in this section, we will use the reduction order method to derive the numerical scheme of an equivalent problem.

Let

and denote

Then Eqs (2.7)–(2.10) are equivalent to

Defining the grid functions on $\Omega_{h\tau}$ as

Considering Eq (3.2a) at the point $(x_{i-\frac{1}{2}}, t_{k-\frac{1}{2}})$ and Eq (3.2b) at the point $(x_{i-\frac{1}{2}}, t_{k})$, and with the help of the Taylor expansion, we have

where $\gamma_{i-\frac{1}{2}}^{k-\frac{1}{2}} = \gamma(x_{i-\frac{1}{2}}, t_{k-\frac{1}{2}})$ and $\theta_{i-\frac{1}{2}}^{k-\frac{1}{2}} = \theta(x_{i-\frac{1}{2}}, t_{k-\frac{1}{2}})$, and there exists a constant $c_1$ such that the local truncation errors satisfy

Omitting the small terms in Eq (3.3a) and Eq (3.3b), a box scheme for Eqs (3.2a)–(3.2d) reads

According to Eq (2.2), we have

There exists a constant $c_3$ such that

Let

We can view $\widehat{u}_{i-\frac{1}{2}}^k$ as the second-order numerical approximation of $u(x_{i-\frac{1}{2}}, t_k)$ according to Eq (3.6).

Theorem 1. The difference scheme Eqs (3.5a)–(3.5d) is equivalent to

and

Proof. First, we know that Eq (3.5b) is equivalent to

Substituting Eq (3.13) into Eq (3.5a), we obtain

Multiplying Eq (3.14) by $\frac{h}{2}$ and adding the result with Eq (3.13), we obtain

Then, multiplying Eq (3.14) by $\frac{h}{2}$ and subtracting the result with Eq (3.13), we obtain

Or equivalently,

It follows from Eq (3.15) and Eq (3.17) with $1 \leqslant i \leqslant M-1$, we get

That is

In Eq (3.17), for $i = 0$, Eq (3.5d) is equivalent to

In Eq (3.15), for $i = M$, Eq (3.5d) is equivalent to

This completes the proof.

3.3.   A prior estimate for the difference scheme

In the following theorem, we use the energy method to give a prior estimate for the difference scheme Eqs (3.5a)–(3.5d).

Theorem 2. Let $\{\tilde{w}_i^k, \tilde{v}_i^k\ | \ 0 \leqslant i \leqslant M, 1 \leqslant k \leqslant N\}$ be the solution of

Then we have

where

Proof. Step 1: Averaging Eq (3.18b) and Eq (3.18c) with superscripts $k$ and $k-1$, we obtain

Multiplying Eq (3.18a) by $2\tilde{w}_{i-\frac{1}{2}}^{k-\frac{1}{2}}$, Eq (3.20) by $2\tilde{v}_{i-\frac{1}{2}}^{k-\frac{1}{2}}$ and adding the results, we have

where $c_2$ is a positive constant.

Multiplying Eq (3.22) by $h$, summing up for $i$ from $1$ to $M$ and noticing Eq (3.21), we obtain

Step 2: Subtracting Eq (3.18b) and Eq (3.18c) with superscripts $k$ and $k-1$, dividing the results by $\tau$ on both sides, we have

Multiplying Eq (3.18a) by $2\delta_t \tilde{w}_{i-\frac{1}{2}}^{k-\frac{1}{2}}$, Eq (3.24) by $2\tilde{v}_{i-\frac{1}{2}}^{k-\frac{1}{2}}$ and adding the results, we obtain

After simplifying Eq (3.26), it becomes

Multiplying Eq (3.27) by $h$, summing up for $i$ from $1$ to $M$ and noticing Eq (3.21), we obtain

Step 3: Adding Eq (3.23) and Eq (3.28), we have

Furthermore, we have

Noticing that Eq (3.19), Eq (3.30) becomes

When $\tau \leqslant \frac{1}{3c_3}$, using Gronwall inequality in Lemma 1 yields

Equivalently,

This completes the proof.

3.4.   Solvability and convergence

Theorem 3 (Solvability). The difference scheme Eqs (3.5a)–(3.5d) is uniquely solvable.

Define

Subtracting Eqs (3.3a)–(3.3d) from Eqs (3.5a)–(3.5d), we have the error system

Noticing that Eqs (3.4a)–(3.4b) and Lemma 2, similar to the proof of the prior estimate in Theorem 2, we have the following convergence results for the above error system.

Theorem 4 (Convergence). Let $\alpha(t)$, $\beta(t) \in C^2[0, T]$, $w(x, t)\in C^4([0, L]\times [0, T])$ and suppose the condition Eq (1.4) is satisfied. Then the solution of the difference scheme Eqs (3.5a)–(3.5d) is convergent to the solution of Eqs (3.2a)–(3.2d) with the order of convergence $O(\tau^2+h^2)$.

4.   Numerical tests

In this section, we will testify to the accuracy and the convergence order for the box scheme Eqs (3.5a)–(3.5d). Based on the discussion in Section 2 and using the equivalence relation in Theorem 1, we give the algorithm flow chart of the box scheme Eqs (3.5a)–(3.5d) for the Burgers' equation Eqs (1.3a)–(1.3c):

Denote

Define the spatial convergence order and temporal convergence order respectively by

Example 1. We first consider the problem with the homogeneous boundary conditions as

The exact solution for the above problem is $u(x, t) = \exp(-t)\sin x$. After simple calculation by Eq (2.8) and Eq (3.1a)and Eq (3.1b), we have $w(x, 0) = 1/(2\mu)\exp(\cos x -1), \; \gamma(x, t) = 0,$ $\; \theta(x, t) = 0.125/\mu \cdot$$\exp(-2t)(\cos 2x-1)-(1-\mu)/\mu \cdot \exp(-t)(\cos x -1)$.

The numerical results are listed in Tables 1, 2 and exact and numerical surfaces are respectively displayed in Figure 1. Table 1 gives the numerical solutions, exact solutions and their absolute errors, which show that the box scheme is efficient even if the coefficient of viscosity is very small. The convergence orders in time and space are displayed in Table 2. We first fix the spatial step size and test the temporal convergence order in the third column and fourth column, which confirm that the temporal convergence order approaches order two. The numerical results in the last two columns verify that the convergence order in spatial convergence order is order two, which is also consistent with our theoretical results. We further compare the numerical solutions and exact solutions in Figure 2, which again demonstrates that the effectiveness of the box scheme whether the coefficient of viscosity is large or small. Moreover, we also notice that the numerical errors increase in a steep fashion, which means that the stiffness of the system intensifies. In these cases, we should use small step sizes to capture the solution profiles accurately.

Example 2. Then we consider the problem with the nonhomogeneous boundary conditions as

The exact solution is $u(x, t) = t^2\cos x$. Here we easily have $w(x, 0) = 1, \ \gamma(x, t) = -0.5/(2\mu) \cdot t^2$, and $\theta(x, t) = 0.25/\mu \cdot t^4 -t/\mu \sin x -0.5t^2 \sin x -0.125/\mu \cdot t^4 \cos 2x +0.125/\mu \cdot t^4$ calculated by using Eq (2.8) and Eq (3.1a) and Eq (3.1b), respectively.

In this example, the boundary conditions and right-hand side terms are both nonhomogeneous. The exact solution surface (left) and numerical solution surface (right) are respectively displayed in Figure 3. The numerical results and error surfaces for different coefficients of viscosity are respectively shown in Tables 3, 4 and Figure 4. Similar results to Example 1 can be observed, which further verify the correctness of our theoretical results.

5.   Conclusion

In summary, it was demonstrated that the Burgers' equation subject to nonhomogeneous Dirichlet boundary conditions is equivalent to a heat equation with derivative boundary conditions based on the Hopf-Cole transformation. We further convert the heat equation into a first-order system with homogeneous boundary conditions via the help of the linear transformation and the reduced-order method. An efficient box scheme is established for the converted first-order system. We further prove that the box scheme is solvable and convergent.

The relationship between the nonlinear convection term $uu_x$ and variable (or constant) coefficient convection term $c(x)u(x, t)$ ($c(x)$ is the variable coefficient) in other references is linked by a generalized exponential transformation. Moreover, the exponential transformation can be applied to solve other partial differential equations with nonlinear convection terms involving fractional differential equations, delay Sobolev equations and delay functional differential equations with Burger-type nonlinear terms ^[26,27]. These will leave to our future research work.

Acknowledgement

The author would like to thank her supervisor Qifeng Zhang, who provide this interesting topic and detailed guidance. The work is supported by Natural Sciences Foundation of Zhejiang Province (Grant No. LZ23A010007).

Conflict of interest

The authors declare there is no conflict of interest.