An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations

Abstract

In this study, we sought numerical solutions for three-dimensional coupled Burgers’ equations. Burgers’ equations are fundamental partial differential equations in fluid mechanics. They integrate the characteristics of both the first-order wave equation and the heat conduction equation, serving as crucial tools for modeling the interaction between convection and diffusion. First, the fractional step method was applied to decompose the equations into one-dimensional forms. Then, implicit finite difference approximations were used to solve the resulting one-dimensional equations. To assess the accuracy of the proposed approach, we tested it on two benchmark problems and compared the results with existing methods in the literature. Additionally, the symmetry of the solution graphs was analyzed to gain deeper insight into the results. Stability analysis using the von Neumann method confirmed that the proposed approach is unconditionally stable. The results obtained in this study strongly support the effectiveness and reliability of the proposed method in solving three-dimensional coupled Burgers’ equations.

1. Introduction

Nonlinear partial differential equations (PDEs) and fractional differential equations (FDEs) play a significant role in modeling various complex phenomena encountered in numerous scientific and engineering disciplines, such as dynamical systems, control engineering, signal processing, and more [1,2,3,4,5,6,7,8,9]. One of these equations is the Burgers’ equation.

Consider the following three-dimensional coupled Burgers’ equations:

$$u_t+u{u}_x+v{u}_y+w{u}_z={\displaystyle \frac{1}{\mathrm{Re}}}\left(u_{xx}+u_{yy}+u_{zz}\right),$$

(1)

$$v_t+u{v}_x+v{v}_y+w{v}_z={\displaystyle \frac{1}{\mathrm{Re}}}\left(v_{xx}+v_{yy}+v_{zz}\right),$$

(2)

$$w_t+u{w}_x+v{w}_y+w{w}_z={\displaystyle \frac{1}{\mathrm{Re}}}\left(w_{xx}+w_{yy}+w_{zz}\right)$$

(3)

with initial conditions:

$$\begin{array}{l}u\left(x, y, z, 0\right)=f_1\left(x, y, z\right), \left(x, y, z\right)\in D,\\ v\left(x, y, z, 0\right)=f_2\left(x, y, z\right), \left(x, y, z\right)\in D,\\ w\left(x, y, z, 0\right)=f_3\left(x, y, z\right), \left(x, y, z\right)\in D\end{array}$$

and boundary conditions:

$$\begin{array}{l}u\left(x, y, z, t\right)=g_1\left(x, y, z, t\right), x, y, z\in \partial D, t\in \left[0, T\right],\\ v\left(x, y, z, t\right)=g_2\left(x, y, z, t\right), x, y, z\in \partial D, t\in \left[0, T\right],\\ w\left(x, y, z, t\right)=g_3\left(x, y, z, t\right), x, y, z\in \partial D, t\in \left[0, T\right]\end{array}$$

where $T$ is the terminal time, $D=\left\{\left(x, y, z\right):a_1\le x\le b_1, a_2\le y\le b_2, a_3\le z\le b_3\right\}$ and $\partial D$ is its boundary, $f_1, f_2, f_3, g_1, g_2$ and $g_3$ are known functions, $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ are the velocity components to be determined, and ${\left(\mathrm{Re}\right)}^{-1}$ is the dimensionless kinematics viscosity where $\mathrm{Re}$ is the Reynolds number.

The coupled viscous Burgers’ equation represents a simplified form of the incompressible Navier–Stokes equation, sharing the same convective and diffusive terms. Numerical solutions to the coupled Burgers’ equation are crucial for fluid flow problems and include hydrodynamic turbulence, shock wave theory, vorticity transport, thermoelastic wave propagation, and dispersion in porous media [10].

In recent years, numerous researchers have employed various numerical methods to solve the Burgers’ equation. While there is extensive literature on the numerical solutions to one- and two-dimensional Burgers’ equations, fewer methods have been proposed for the three-dimensional coupled Burgers’ equation. Jain and Holla [11] proposed two algorithms based on the cubic spline function technique for obtaining numerical solutions to one- and two-dimensional Burgers’ equations. Fletcher [12] studied numerical solutions to one- and two-dimensional Burgers’ equations using finite element and finite difference methods and compared the results obtained by the two methods. Wubs and Goede [13] used an explicit and implicit method. Goyon [14] used a variety of multilevel approaches to solve the equation. Bahadır [15] obtained numerical solutions to the system of equations by using the fully implicit finite difference method. El-Sayed and Kaya [16] obtained numerical solutions to the system of two-dimensional Burgers’ equations by using the decomposition method. Abdou and Soliman [17] used the variational iteration method to solve the two-dimensional Burgers’ equation. Celikten and Aksan [18] used the alternating direction implicit method to solve the system of two-dimensional Burgers’ equations. Li et al. [19] proposed an efficient localized meshless collocation method for numerical solutions to two-dimensional Burgers-type equations. Yousif and Hamasalh [20,21] used the conformable non-polynomial spline method and a non-polynomial spline fractional continuity method to solve the time-fractional Burgers–Fisher equation. Çelikten and Cankurt [22] employed a Crank Nicolson logarithmic finite difference method to obtain numerical solutions to the generalized Burgers–Huxley equation. Srivastava et al. [23] proposed a general analytical solution for the three-dimensional (3D) homogeneous, coupled, unsteady, nonlinear, generalized viscous Burgers’ equation using the Hopf–Cole transformation and the separation of variables methods. Rezaei and Vaghefi [24] employed a combined differential quadrature and finite difference method (DQ-FDM) to obtain numerical solutions to the 3D Burgers’ equation, determining the three-dimensional velocity field. Shukla et al. [10] introduced a modified cubic B-spline differential quadrature method (MCB-DQM) to solve the 3D coupled viscous Burgers’ equation with appropriate initial and boundary conditions. Alhendi and Alderremy [25] utilized several numerical approaches, including the Laplace transform–Adomian decomposition method (LT-ADM), the Laplace transform homotopy perturbation method (LT-HPM), the variational iteration method (VIM), the variational iteration decomposition method (VIDM), and the variational iteration homotopy perturbation method (VIHPM) to solve the three-dimensional coupled Burgers’ equation. G. Singh and I. Singh [26] applied the new Laplace variational iteration method (NLVIM), which combines the Laplace transform with a modified variational iteration technique, to solve the three-dimensional (3D) coupled Burgers’ equation.

The motivation for this study arises from the inherent challenges associated with high-dimensional nonlinear differential equations and the need for accurate and efficient numerical techniques to solve them. Traditional methods often fail to effectively capture the complex behavior of such equations, making analytical solutions difficult to obtain. Consequently, numerical methods have become indispensable tools for approximating solutions. In this context, the fractional step method presents a promising approach due to its ability to decompose high-dimensional equations into one-dimensional forms, thereby simplifying computations while preserving accuracy. The proposed method offers several advantages, including precise solution approximation, stability, and reduced computational cost without compromising accuracy. Furthermore, it demonstrates superior performance compared to existing methods in terms of error reduction and computational efficiency. This study introduces a robust and accurate numerical approach based on the fractional step method, which decomposes the three-dimensional coupled Burgers’ equation into one-dimensional forms. These reduced forms are then solved using implicit finite difference approximations. To evaluate the accuracy and efficiency of the proposed method, two benchmark problems are examined. The obtained results are compared with exact solutions and numerical findings from previous studies in the literature. The results confirm that the proposed method provides accurate and reliable solutions, demonstrating consistency with those obtained in other studies.

The remaining sections of the paper are structured as follows:

Section 2: The numerical method for solving the three-dimensional (3D) coupled Burgers’ equation is defined and a stability analysis for the method is presented;

Section 3: Numerical results are presented to assess the effectiveness of the method, illustrated through two problems;

Section 4: Conclusions are drawn, and some comments for future work are provided.

2. Materials and Methods

2.1. Fractional Step Method

Fractional step methods decompose multidimensional equations into a series of one-dimensional equations [27], which are then approximated using either implicit or explicit schemes [28]. In this study, to obtain numerical solutions to three-dimensional coupled Burgers’ equations, we first applied the fractional step method to split the equations into one-dimensional equations, and then, we used the implicit finite difference approximations to solve the obtained one-dimensional equations. The key advantages of the proposed numerical scheme include the following:

• Fractional step decomposition—The three-dimensional coupled Burgers’ equations are efficiently transformed into simpler one-dimensional forms, reducing computational effort;
• Implicit finite difference scheme—Unlike explicit methods, the implicit approach enhances numerical stability and accuracy;
• Unconditional stability—A rigorous von Neumann stability analysis proves that the method is unconditionally stable, ensuring reliable numerical solutions;
• Benchmark validation—The method is tested on standard benchmark problems, demonstrating superior accuracy and robustness compared to existing techniques;
• Computational efficiency—The decomposition strategy reduces computational costs while maintaining high accuracy, making the approach suitable for large-scale simulations in fluid dynamics and related fields.

We indicate the discrete approximation of $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ at grid point $\left(x_i, y_j, z_p, t_n\right)$ by $u_{i,j,p}^n$, $v_{i,j,p}^n$ and $w_{i,j,p}^n$, respectively, $\left(i=0, 1, 2, \dots , N_x; j=0, 1, 2, \dots , N_y; p=0, 1, 2, \dots , N_z; n=0, 1, 2, \dots , M\right)$ where $h_x={\displaystyle \frac{b_1-a_1}{N_x}}$, $h_y={\displaystyle \frac{b_2-a_2}{N_y}}$, and $h_z={\displaystyle \frac{b_3-a_3}{N_z}}$ are grid sizes in x-direction, y-direction, and z-direction, respectively, and $k={\displaystyle \frac{T}{M}}$ represents the increment in time. The flowchart of the proposed method is provided in Figure 1.

Therefore, by applying the fractional step method to Equation (1), we decompose it into the following three one-dimensional partial differential equations, which are considered instead of Equation (1):

$$\begin{array}{l}{\displaystyle \frac{u_t}{3}}+u{u}_x={\displaystyle \frac{u_{xx}}{\mathrm{Re}}},\\ {\displaystyle \frac{u_t}{3}}+v{u}_y={\displaystyle \frac{u_{yy}}{\mathrm{Re}}},\\ {\displaystyle \frac{u_t}{3}}+w{u}_z={\displaystyle \frac{u_{zz}}{\mathrm{Re}}}.\end{array}$$

(4)

Similarly, we consider the following three one-dimensional partial differential equations instead of Equation (2):

$$\begin{array}{l}{\displaystyle \frac{v_t}{3}}+u{v}_x={\displaystyle \frac{v_{xx}}{\mathrm{Re}}},\\ {\displaystyle \frac{v_t}{3}}+v{v}_y={\displaystyle \frac{v_{yy}}{\mathrm{Re}}},\\ {\displaystyle \frac{v_t}{3}}+w{v}_z={\displaystyle \frac{v_{zz}}{\mathrm{Re}}}\end{array}$$

(5)

Moreover, we consider the following three one-dimensional partial differential equations instead of Equation (3):

$$\begin{array}{l}{\displaystyle \frac{w_t}{3}}+u{w}_x={\displaystyle \frac{w_{xx}}{\mathrm{Re}}},\\ {\displaystyle \frac{w_t}{3}}+v{w}_y={\displaystyle \frac{w_{yy}}{\mathrm{Re}}},\\ {\displaystyle \frac{w_t}{3}}+w{w}_z={\displaystyle \frac{w_{zz}}{\mathrm{Re}}}.\end{array}$$

(6)

The implicit finite difference approximations that we propose to Equation (4) are in the following form:

$${\displaystyle \frac{u_{i,j,p}^{n+\frac{1}{3}}-u_{i,j,p}^n}{k}}+u_{i,j,p}^n\left({\displaystyle \frac{u_{i+1,j,p}^{n+\frac{1}{3}}-u_{i-1,j,p}^{n+\frac{1}{3}}}{2h_x}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{u_{i+1,j,p}^{n+\frac{1}{3}}-2u_{i,j,p}^{n+\frac{1}{3}}+u_{i-1,j,p}^{n+\frac{1}{3}}}{h_x^2}}\right),$$

(7)

$${\displaystyle \frac{u_{i,j,p}^{n+\frac{2}{3}}-u_{i,j,p}^{n+\frac{1}{3}}}{k}}+v_{i,j,p}^{n+\frac{1}{3}}\left({\displaystyle \frac{u_{i,j+1,p}^{n+\frac{2}{3}}-u_{i,j-1,p}^{n+\frac{2}{3}}}{2h_y}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{u_{i,j+1p}^{n+\frac{2}{3}}-2u_{i,j,p}^{n+\frac{2}{3}}+u_{i,j-1,p}^{n+\frac{2}{3}}}{h_y^2}}\right),$$

(8)

$${\displaystyle \frac{u_{i,j,p}^{n+1}-u_{i,j,p}^{n+\frac{2}{3}}}{k}}+w_{i,j,p}^{n+\frac{2}{3}}\left({\displaystyle \frac{u_{i,j,p+1}^{n+1}-u_{i,j,p-1}^{n+1}}{2h_z}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{u_{i,j,p+1}^{n+1}-2u_{i,j,p}^{n+1}+u_{i,j,p-1}^{n+1}}{h_z^2}}\right).$$

(9)

Similarly, the implicit finite difference approximations that we propose to Equation (5) are in the following form:

$${\displaystyle \frac{v_{i,j,p}^{n+\frac{1}{3}}-v_{i,j,p}^n}{k}}+u_{i,j,p}^n\left({\displaystyle \frac{v_{i+1,j,p}^{n+\frac{1}{3}}-v_{i-1,j,p}^{n+\frac{1}{3}}}{2h_x}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{v_{i+1,j,p}^{n+\frac{1}{3}}-2v_{i,j,p}^{n+\frac{1}{3}}+v_{i-1,j,p}^{n+\frac{1}{3}}}{h_x^2}}\right),$$

(10)

$${\displaystyle \frac{v_{i,j,p}^{n+\frac{2}{3}}-v_{i,j,p}^{n+\frac{1}{3}}}{k}}+v_{i,j,p}^{n+\frac{1}{3}}\left({\displaystyle \frac{v_{i,j+1,p}^{n+\frac{2}{3}}-v_{i,j-1,p}^{n+\frac{2}{3}}}{2h_y}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{v_{i,j+1p}^{n+\frac{2}{3}}-2v_{i,j,p}^{n+\frac{2}{3}}+v_{i,j-1,p}^{n+\frac{2}{3}}}{h_y^2}}\right),$$

(11)

$${\displaystyle \frac{v_{i,j,p}^{n+1}-v_{i,j,p}^{n+\frac{2}{3}}}{k}}+w_{i,j,p}^{n+\frac{2}{3}}\left({\displaystyle \frac{v_{i,j,p+1}^{n+1}-v_{i,j,p-1}^{n+1}}{2h_z}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{v_{i,j,p+1}^{n+1}-2v_{i,j,p}^{n+1}+v_{i,j,p-1}^{n+1}}{h_z^2}}\right)$$

(12)

Moreover, the implicit finite difference approximations that we propose to Equation (6) are in the following form:

$${\displaystyle \frac{w_{i,j,p}^{n+\frac{1}{3}}-w_{i,j,p}^n}{k}}+u_{i,j,p}^n\left({\displaystyle \frac{w_{i+1,j,p}^{n+\frac{1}{3}}-w_{i-1,j,p}^{n+\frac{1}{3}}}{2h_x}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{w_{i+1,j,p}^{n+\frac{1}{3}}-2w_{i,j,p}^{n+\frac{1}{3}}+w_{i-1,j,p}^{n+\frac{1}{3}}}{h_x^2}}\right),$$

(13)

$${\displaystyle \frac{w_{i,j,p}^{n+\frac{2}{3}}-w_{i,j,p}^{n+\frac{1}{3}}}{k}}+v_{i,j,p}^{n+\frac{1}{3}}\left({\displaystyle \frac{w_{i,j+1,p}^{n+\frac{2}{3}}-w_{i,j-1,p}^{n+\frac{2}{3}}}{2h_y}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{w_{i,j+1p}^{n+\frac{2}{3}}-2w_{i,j,p}^{n+\frac{2}{3}}+w_{i,j-1,p}^{n+\frac{2}{3}}}{h_y^2}}\right),$$

(14)

$${\displaystyle \frac{w_{i,j,p}^{n+1}-w_{i,j,p}^{n+\frac{2}{3}}}{k}}+w_{i,j,p}^{n+\frac{2}{3}}\left({\displaystyle \frac{w_{i,j,p+1}^{n+1}-w_{i,j,p-1}^{n+1}}{2h_z}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{w_{i,j,p+1}^{n+1}-2w_{i,j,p}^{n+1}+w_{i,j,p-1}^{n+1}}{h_z^2}}\right).$$

(15)

For simplicity, we let $h_x=h_y=h_z=h$; hence, (7)–(9) approximations are given in the following form:

$$\left(-r_1{u}_{i,j,p}^n-r_2\right)u_{i-1,j,p}^{n+\frac{1}{3}}+\left(1+2r_2\right)u_{i,j,p}^{n+\frac{1}{3}}+\left(r_1{u}_{i,j,p}^n-r_2\right)u_{i+1,j,p}^{n+\frac{1}{3}}=u_{i,j,p}^n,$$

(16)

$$\left(-r_1{v}_{i,j,p}^{n+\frac{1}{3}}-r_2\right)u_{i,j-1,p}^{n+\frac{2}{3}}+\left(1+2r_2\right)u_{i,j,p}^{n+\frac{2}{3}}+\left(r_1{v}_{i,j,p}^{n+\frac{1}{3}}-r_2\right)u_{i,j+1,p}^{n+\frac{2}{3}}=u_{i,j,p}^{n+\frac{1}{3}},$$

(17)

$$\left(-r_1{w}_{i,j,p}^{n+\frac{2}{3}}-r_2\right)u_{i,j,p-1}^{n+1}+\left(1+2r_2\right)u_{i,j,p}^{n+1}+\left(r_1{w}_{i,j,p}^{n+\frac{2}{3}}-r_2\right)u_{i,j,p+1}^{n+1}=u_{i,j,p}^{n+\frac{2}{3}},$$

(18)

where $r_1={\displaystyle \frac{k}{2h}}$ and $r_2={\displaystyle \frac{k}{\mathrm{Re}{h}^2}}$.

Similarly, (10)–(12) approximations are given in the following form:

$$\left(-r_1{u}_{i,j,p}^n-r_2\right)v_{i-1,j,p}^{n+\frac{1}{3}}+\left(1+2r_2\right)v_{i,j,p}^{n+\frac{1}{3}}+\left(r_1{u}_{i,j,p}^n-r_2\right)v_{i+1,j,p}^{n+\frac{1}{3}}=v_{i,j,p}^n,$$

(19)

$$\left(-r_1{v}_{i,j,p}^{n+\frac{1}{3}}-r_2\right)v_{i,j-1,p}^{n+\frac{2}{3}}+\left(1+2r_2\right)v_{i,j,p}^{n+\frac{2}{3}}+\left(r_1{v}_{i,j,p}^{n+\frac{1}{3}}-r_2\right)v_{i,j+1,p}^{n+\frac{2}{3}}=v_{i,j,p}^{n+\frac{1}{3}},$$

(20)

$$\left(-r_1{w}_{i,j,p}^{n+\frac{2}{3}}-r_2\right)v_{i,j,p-1}^{n+1}+\left(1+2r_2\right)v_{i,j,p}^{n+1}+\left(r_1{w}_{i,j,p}^{n+\frac{2}{3}}-r_2\right)v_{i,j,p+1}^{n+1}=v_{i,j,p}^{n+\frac{2}{3}}.$$

(21)

Moreover, (13)–(15) approximations are given in the following form:

$$\left(-r_1{u}_{i,j,p}^n-r_2\right)w_{i-1,j,p}^{n+\frac{1}{3}}+\left(1+2r_2\right)w_{i,j,p}^{n+\frac{1}{3}}+\left(r_1{u}_{i,j,p}^n-r_2\right)w_{i+1,j,p}^{n+\frac{1}{3}}=w_{i,j,p}^n,$$

(22)

$$\left(-r_1{v}_{i,j,p}^{n+\frac{1}{3}}-r_2\right)w_{i,j-1,p}^{n+\frac{2}{3}}+\left(1+2r_2\right)w_{i,j,p}^{n+\frac{2}{3}}+\left(r_1{v}_{i,j,p}^{n+\frac{1}{3}}-r_2\right)w_{i,j+1,p}^{n+\frac{2}{3}}=w_{i,j,p}^{n+\frac{1}{3}},$$

(23)

$$\left(-r_1{w}_{i,j,p}^{n+\frac{2}{3}}-r_2\right)w_{i,j,p-1}^{n+1}+\left(1+2r_2\right)w_{i,j,p}^{n+1}+\left(r_1{w}_{i,j,p}^{n+\frac{2}{3}}-r_2\right)w_{i,j,p+1}^{n+1}=w_{i,j,p}^{n+\frac{2}{3}},$$

(24)

2.2. Local Truncation Error (LTE) and Consistency

To analyze the consistency of the numerical schemes, the nonlinear terms $u{u}_x$, $v{u}_y$, and $w{u}_z$ in three-dimensional Burgers’ Equation (1) have been linearized by replacing the quantities $u$, $v$, and $w$ with local content. Thus, the nonlinear terms $u{u}_x$, $v{u}_y$, and $w{u}_z$ convert into $\widehat{U}{u}_x$, $\widehat{V}{u}_y$, and $\tilde{W}{u}_z$ in Equation (1), and in that case, Equation (1) turns into

$$u_t+\widehat{U}{u}_x+\widehat{V}{u}_y+\tilde{W}{u}_z={\displaystyle \frac{1}{\mathrm{Re}}}\left(u_{xx}+u_{yy}+u_{zz}\right).$$

(25)

If we write the numerical schemes (7)–(9) for this linearized equation, the new approximations become the following:

$${\displaystyle \frac{u_{i,j,p}^{n+\frac{1}{3}}-u_{i,j,p}^n}{k}}+\tilde{U}\left({\displaystyle \frac{u_{i+1,j,p}^{n+\frac{1}{3}}-u_{i-1,j,p}^{n+\frac{1}{3}}}{2h_x}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{u_{i+1,j,p}^{n+\frac{1}{3}}-2u_{i,j,p}^{n+\frac{1}{3}}+u_{i-1,j,p}^{n+\frac{1}{3}}}{h_x^2}}\right),$$

(26)

$${\displaystyle \frac{u_{i,j,p}^{n+\frac{2}{3}}-u_{i,j,p}^{n+\frac{1}{3}}}{k}}+\tilde{V}\left({\displaystyle \frac{u_{i,j+1,p}^{n+\frac{2}{3}}-u_{i,j-1,p}^{n+\frac{2}{3}}}{2h_y}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{u_{i,j+1p}^{n+\frac{2}{3}}-2u_{i,j,p}^{n+\frac{2}{3}}+u_{i,j-1,p}^{n+\frac{2}{3}}}{h_y^2}}\right),$$

(27)

$${\displaystyle \frac{u_{i,j,p}^{n+1}-u_{i,j,p}^{n+\frac{2}{3}}}{k}}+\tilde{W}\left({\displaystyle \frac{u_{i,j,p+1}^{n+1}-u_{i,j,p-1}^{n+1}}{2h_z}}\right)={\displaystyle \frac{1}{\mathrm{Re}}}\left({\displaystyle \frac{u_{i,j,p+1}^{n+1}-2u_{i,j,p}^{n+1}+u_{i,j,p-1}^{n+1}}{h_z^2}}\right).$$

(28)

In order to prove that the schemes given by (26)–(28) are consistent, we will show the local truncation error of the schemes. If we sum the schemes given by (26)–(28) and then use the Taylor series of $\left(x_i, y_j, z_p, t_n\right)$, we obtain the local truncation error as follows:

$$\begin{array}{l}{T}_{i, j, p}^n={\left[u_t+\widehat{U}{u}_x+\widehat{V}{u}_y+\tilde{W}{u}_z={\displaystyle \frac{1}{\mathrm{Re}}}\left(u_{xx}+u_{yy}+u_{zz}\right)\right]}_{i, j, \mathrm{p}}^n\\ +{\displaystyle \frac{k}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\tilde{U}{\displaystyle \frac{h_x^2}{3}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\tilde{V}{\displaystyle \frac{h_y^2}{3}}{\displaystyle \frac{{\partial }^{3}u}{\partial y^3}}+\tilde{W}{\displaystyle \frac{h_z^2}{3}}{\displaystyle \frac{{\partial }^{3}u}{\partial z^3}}+\cdots \end{array}$$

Thus, the principal part of the local truncation error is as follows:

$${\left[{\displaystyle \frac{k}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\tilde{U}{\displaystyle \frac{h_x^2}{3}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\tilde{V}{\displaystyle \frac{h_y^2}{3}}{\displaystyle \frac{{\partial }^{3}u}{\partial y^3}}+\tilde{W}{\displaystyle \frac{h_z^2}{3}}{\displaystyle \frac{{\partial }^{3}u}{\partial z^3}}\right]}_{i, j, p}^n.$$

So, the local truncation error (LTE) is $O\left(h_x^2+h_y^2+h_z^2+k\right)$.

The consistency of a finite difference scheme refers to how well the numerical approximation represents the original equation. A numerical scheme is consistent if the local truncation error (LTE) tends toward zero as the step sizes $h_x, h_y, h_z, k$ approach zero. Since $\underset{h_x, h_y, h_z, k\to 0}{\lim }LTE=0$, the schemes (7)–(9) are consistent.

Similarly, we can show the consistency of the numerical schemes (10)–(12) and numerical schemes (13)–(15).

2.3. Stability Analysis

We use the von Neumann stability analysis to prove the unconditional stability of the schemes obtained using the fractional step method. Let

$$u_{i,j,p}^n={\gamma }^n{e}^{I\alpha ih}{e}^{I\beta jh}{e}^{I\eta ph}$$

(29)

where $I=\sqrt{-1}$.

To analyze the stability of the numerical schemes, we consider linearized Equation (26). If we split linearized Equation (26) by the fractional step method, we then use the new implicit finite difference approximation schemes as follows:

$$\left(-r_1\tilde{U}-r_2\right)u_{i-1,j,p}^{n+\frac{1}{3}}+\left(1+2r_2\right)u_{i,j,p}^{n+\frac{1}{3}}+\left(r_1\tilde{U}-r_2\right)u_{i+1,j,p}^{n+\frac{1}{3}}=u_{i,j,p}^n,$$

(30)

$$\left(-r_1\tilde{V}-r_2\right)u_{i,j-1,p}^{n+\frac{2}{3}}+\left(1+2r_2\right)u_{i,j,p}^{n+\frac{2}{3}}+\left(r_1\tilde{V}-r_2\right)u_{i,j+1,p}^{n+\frac{2}{3}}=u_{i,j,p}^{n+\frac{1}{3}},$$

(31)

$$\left(-r_1\tilde{W}-r_2\right)u_{i,j,p-1}^{n+1}+\left(1+2r_2\right)u_{i,j,p}^{n+1}+\left(r_1\tilde{W}-r_2\right)u_{i,j,p+1}^{n+1}=u_{i,j,p}^{n+\frac{2}{3}}.$$

(32)

Then, substituting Equation (29) into the new numerical schemes (30)–(32), we obtain the following:

$${\gamma }^{n+\frac{1}{3}}\left(1+2r_2+r_1\tilde{U}\left(e^{I\alpha h}-e^{-I\alpha h}\right)-r_2\left(e^{I\alpha h}-e^{-I\alpha h}\right)\right)={\gamma }^n,$$

$${\gamma }^{n+\frac{2}{3}}\left(1+2r_2+r_1\tilde{V}\left(e^{I\beta h}-e^{-I\beta h}\right)-r_2\left(e^{I\beta h}-e^{-I\beta h}\right)\right)={\gamma }^{n+\frac{1}{3}},$$

$${\gamma }^{n+1}\left(1+2r_2+r_1\tilde{W}\left(e^{I\eta h}-e^{-I\eta h}\right)-r_2\left(e^{I\eta h}-e^{-I\eta h}\right)\right)={\gamma }^{n+\frac{2}{3}},$$

After making specific simplifications and consolidating pertinent terms, in a scenario where $\cos \left(\alpha h\right)={\displaystyle \frac{e^{I\alpha h}+e^{-I\alpha h}}{2}}$ and $\sin \left(\alpha h\right)={\displaystyle \frac{e^{I\alpha h}-e^{-I\alpha h}}{2I}}$, the resultant expressions become the following:

$${\gamma }^{n+\frac{1}{3}}\left(1+2r_2(1-\cos (\alpha h))+2r_1\tilde{U}\sin (\alpha h)I\right)={\gamma }^n,$$

$${\gamma }^{n+\frac{2}{3}}\left(1+2r_2(1-\cos (\beta h))+2r_1\tilde{V}\sin (\beta h)I\right)={\gamma }^{n+\frac{1}{3}},$$

$${\gamma }^{n+1}\left(1+2r_2(1-\cos (\eta h)+2r_1\tilde{W}\sin (\eta h)I\right)={\gamma }^{n+\frac{2}{3}}.$$

Then, using equation $1-\cos (\alpha h)={\sin }^2\left({\displaystyle \frac{\alpha h}{2}}\right)$, we get the following growth factors:

$$\begin{array}{l}{\displaystyle \frac{{\gamma }^{n+\frac{1}{3}}}{{\gamma }^n}}={\displaystyle \frac{1}{1+2r_2{\sin }^2\frac{\alpha h}{2}+\left(2r_1\tilde{U}\sin \alpha h\right)I}},\\ {\displaystyle \frac{{\gamma }^{n+\frac{2}{3}}}{{\gamma }^{n+\frac{1}{3}}}}={\displaystyle \frac{1}{1+2r_2{\sin }^2\frac{\beta h}{2}+\left(2r_1\tilde{V}\sin \beta h\right)I}},\\ {\displaystyle \frac{{\gamma }^{n+1}}{{\gamma }^{n+\frac{2}{3}}}}={\displaystyle \frac{1}{1+2r_2{\sin }^2\frac{\eta h}{2}+\left(2r_1\tilde{W}\sin \eta h\right)I}}.\end{array}$$

Hence,

$${\displaystyle \frac{{\gamma }^{n+1}}{{\gamma }^n}}=\left({\displaystyle \frac{1}{1+{\lambda }_1}}\right)\left({\displaystyle \frac{1}{1+{\lambda }_2}}\right)\left({\displaystyle \frac{1}{1+{\lambda }_3}}\right),$$

where ${\lambda }_1=2r_2{\sin }^2{\displaystyle \frac{\alpha h}{2}}+\left(2r_1\tilde{U}\sin \alpha h\right)I, {\lambda }_2=2r_2{\sin }^2{\displaystyle \frac{\beta h}{2}}+\left(2r_1\tilde{V}\sin \beta h\right)I$ and ${\lambda }_3=2r_2{\sin }^2{\displaystyle \frac{\eta h}{2}}+\left(2r_1\tilde{W}\sin \eta h\right)I$.

The growth factor from $n$ to $n+1$ must be less than 1 in absolute value for unconditional stability.

Since $r_1,r_2>0$, it can be seen that $\left|{\displaystyle \frac{{\gamma }^{n+1}}{{\gamma }^n}}\right|\le 1$. Hence, schemes (16)–(18) are unconditionally stable.

Similarly, we can demonstrate the unconditional stability of the numerical schemes (19)–(21) and (22)–(24). In fact, each scheme is independently stable as well.

2.4. Convergence

According to Lax’s equivalence theorem, a consistent finite difference equation is convergent if it is stable. The proposed approximations for Equation (1) have been shown to be both consistent and stable. Therefore, by Lax’s equivalence theorem, the proposed approximations for Equation (1) are convergent.

Similarly, it can be shown that the proposed approximations for Equations (2) and (3) are also convergent.

3. Results

To illustrate the accuracy and efficiency of the present method, two model problems are investigated. To demonstrate the correctness of the results, $L_2$ and $L_{\infty }$ error norms and absolute error are used as follows:

$$L_2={‖U-u‖}_2={\left({\displaystyle \sum _{i=0}^{N_x}{\displaystyle \sum _{j=0}^{N_y}{\displaystyle \sum _{p=0}^{N_z}{\left|U_{i,j,p}-u_{i,j,p}\right|}^2}}}\right)}^{{\displaystyle \frac{1}{2}}},$$

$$L_{\infty }={‖U-u‖}_{\infty }=\underset{i,j}{\max }\left|U_{i,j,p}-u_{i,j,p}\right|,$$

$$A.E.=\left|U-u\right|=\left|U_{i,j,p}-u_{i,j,p}\right|$$

where $U$ and $u$ indicate exact and computed numerical solutions, respectively.

Problem 1: We consider the three-dimensional coupled Burgers’ Equations (1)–(3) with exact solutions:

$$\begin{array}{l}u\left(x, y, z, t\right)=-{\displaystyle \frac{2}{\mathrm{Re}}}\left({\displaystyle \frac{1+\cos \left(x\right)\sin \left(y\right)\sin \left(z\right)\exp (-t)}{1+x+\sin \left(x\right)\sin \left(y\right)\sin \left(z\right)\exp (-t)}}\right) ,\\ v\left(x, y, z, t\right)=-{\displaystyle \frac{2}{\mathrm{Re}}}\left({\displaystyle \frac{\sin \left(x\right)\cos \left(y\right)\sin \left(z\right)\exp (-t)}{1+x+\sin \left(x\right)\sin \left(y\right)\sin \left(z\right)\exp (-t)}}\right) ,\\ w\left(x, y, z, t\right)=-{\displaystyle \frac{2}{\mathrm{Re}}}\left({\displaystyle \frac{\sin \left(x\right)\sin \left(y\right)\cos \left(z\right)\exp (-t)}{1+x+\sin \left(x\right)\sin \left(y\right)\sin \left(z\right)\exp (-t)}}\right)\end{array}$$

which can be generated by using the Hopf–Cole transformation [1]. The initial and boundary conditions are taken from the exact solutions, and the computational domain is $D=\left\{\left(x, y, z, t\right):0\le x\le 1, 0\le y\le 1, 0\le z\le 1\right\}$.

We used the mesh width $h_x=h_y=h_z=h=0.01$, $k=0.0001$, and $\mathrm{Re}=100$ for

Table 1. Comparison of numerical solutions and exact solutions to $u\left(x, y, z, t\right)$.

$\mathit{t}$	$\left(\mathit{x}, \mathit{y}, \mathit{z}\right)$	Exact	Present	Absolute Error
0.002	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.01834520 −0.01896376 −0.01945593 −0.01340710 −0.01368335 −0.01389983 −0.01492331 −0.01582336 −0.01054814 −0.01062954 −0.01098810 −0.01124165 −0.01160368	−0.01834552 −0.01896527 −0.01945839 −0.01340724 −0.01368402 −0.01390090 −0.01492619 −0.01582768 −0.01054819 −0.01062973 −0.01098892 −0.01124285 −0.01160535	$0.03169903\times {10}^{-5}$ $0.15118712\times {10}^{-5}$ $0.24567504\times {10}^{-5}$ $0.01427884\times {10}^{-5}$ $0.06694601\times {10}^{-5}$ $0.10732443\times {10}^{-5}$ $0.28762698\times {10}^{-5}$ $0.43171836\times {10}^{-5}$ $0.00422145\times {10}^{-5}$ $0.01965668\times {10}^{-5}$ $0.08191225\times {10}^{-5}$ $0.12028640\times {10}^{-5}$ $0.16697475\times {10}^{-5}$
0.006	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.01834455 −0.01896065 −0.01945088 −0.01340680 −0.01368198 −0.01389762 −0.01491740 −0.01581448 −0.01054806 −0.01062913 −0.01098642 −0.01123917 −0.01160025	−0.01834550 −0.01896518 −0.01945824 −0.01340723 −0.01368398 −0.01390084 −0.01492601 −0.01582741 −0.01054818 −0.01062972 −0.01098887 −0.01124278 −0.01160525	$0.09490187\times {10}^{-5}$ $0.45263656\times {10}^{-5}$ $0.73553059\times {10}^{-5}$ $0.04274899\times {10}^{-5}$ $0.20043737\times {10}^{-5}$ $0.32134324\times {10}^{-5}$ $0.86134823\times {10}^{-5}$ $1.29306021\times {10}^{-5}$ $0.01263853\times {10}^{-5}$ $0.05885342\times {10}^{-5}$ $0.24531853\times {10}^{-5}$ $0.36031534\times {10}^{-5}$ $0.50030873\times {10}^{-5}$
0.010	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.01834390 −0.01895756 −0.01944585 −0.01340651 −0.01368061 −0.01389542 −0.01491150 −0.01580563 −0.01054797 −0.01062873 −0.01098474 −0.01123671 −0.01159682	−0.01834548 −0.01896508 −0.01945809 −0.01340722 −0.01368394 −0.01390077 −0.01492583 −0.01582715 −0.01054818 −0.01062971 −0.01098882 −0.01124270 −0.01160515	$0.15784527\times {10}^{-5}$ $0.75285699\times {10}^{-5}$ $1.22339985\times {10}^{-5}$ $0.07110279\times {10}^{-5}$ $0.33339618\times {10}^{-5}$ $0.53452455\times {10}^{-5}$ $1.43303076\times {10}^{-5}$ $2.15161394\times {10}^{-5}$ $0.02102127\times {10}^{-5}$ $0.09789517\times {10}^{-5}$ $0.40816833\times {10}^{-5}$ $0.59962013\times {10}^{-5}$ $0.83282216\times {10}^{-5}$

,

Table 2. Comparison of numerical solutions and exact solutions to $v\left(x, y, z, t\right)$.

$\mathit{t}$	$\left(\mathit{x}, \mathit{y}, \mathit{z}\right)$	Exact	Present	Absolute Error
0.002	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.00017979 −0.00086042 −0.00140200 −0.00063170 −0.00299749 −0.00485131 −0.00250076 −0.00391638 −0.00081409 −0.00384970 −0.00316312 −0.00212417 −0.00319923	−0.00018013 −0.00086209 −0.00140470 −0.00063293 −0.00300322 −0.00486050 −0.00250529 −0.00392317 −0.00081567 −0.00385704 −0.00316873 −0.00212774 −0.00320419	$0.03488062\times {10}^{-5}$ $0.16636154\times {10}^{-5}$ $0.27033307\times {10}^{-5}$ $0.12228008\times {10}^{-5}$ $0.57330695\times {10}^{-5}$ $0.91909614\times {10}^{-5}$ $0.45238704\times {10}^{-5}$ $0.67901757\times {10}^{-5}$ $0.15744064\times {10}^{-5}$ $0.73310251\times {10}^{-5}$ $0.56107579\times {10}^{-5}$ $0.35718832\times {10}^{-5}$ $0.49582843\times {10}^{-5}$
0.006	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.00017907 −0.00085700 −0.00139644 −0.00062919 −0.00298570 −0.00483241 −0.00249146 −0.00390242 −0.00081086 −0.00383463 −0.00315158 −0.00211682 −0.00318902	−0.00018011 −0.00086198 −0.00140453 −0.00063285 −0.00300286 −0.00485993 −0.00250501 −0.00392275 −0.00081557 −0.00385658 −0.00316838 −0.00212752 −0.00320388	$0.10442703\times {10}^{-5}$ $0.49806701\times {10}^{-5}$ $0.80935467\times {10}^{-5}$ $0.36609054\times {10}^{-5}$ $1.71648969\times {10}^{-5}$ $2.75189301\times {10}^{-5}$ $1.35475044\times {10}^{-5}$ $2.03375784\times {10}^{-5}$ $0.47135843\times {10}^{-5}$ $2.19495862\times {10}^{-5}$ $1.68036255\times {10}^{-5}$ $1.06994996\times {10}^{-5}$ $1.48565750\times {10}^{-5}$
0.010	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.00017835 −0.00085360 −0.00139090 −0.00062669 −0.00297396 −0.00481359 −0.00248219 −0.00388849 −0.00080763 −0.00381962 −0.00314008 −0.00210949 −0.00317884	−0.00018009 −0.00086188 −0.00140437 −0.00063278 −0.00300251 −0.00485936 −0.00250473 −0.00392233 −0.00081547 −0.00385613 −0.00316804 −0.00212730 −0.00320357	$0.17368796\times {10}^{-5}$ $0.82842012\times {10}^{-5}$ $1.34619062\times {10}^{-5}$ $0.60890470\times {10}^{-5}$ $2.85511187\times {10}^{-5}$ $4.57751758\times {10}^{-5}$ $2.25390728\times {10}^{-5}$ $3.38411290\times {10}^{-5}$ $0.78399576\times {10}^{-5}$ $3.65103395\times {10}^{-5}$ $2.79583768\times {10}^{-5}$ $1.78056126\times {10}^{-5}$ $2.47304995\times {10}^{-5}$

,

Table 3. Comparison of numerical solutions and exact solutions to $w\left(x, y, z, t\right)$.

$\mathit{t}$	$\left(\mathit{x}, \mathit{y}, \mathit{z}\right)$	Exact	Present	Absolute Error
0.002	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.00017979 −0.00015803 −0.00011163 −0.00063170 −0.00055052 −0.00038626 −0.00250076 −0.00169783 −0.00081409 −0.00070704 −0.00316312 −0.00489982 −0.00319923	−0.00018013 −0.00015833 −0.00011184 −0.00063293 −0.00055158 −0.00038700 −0.00250529 −0.00170077 −0.00081567 −0.00070839 −0.00316873 −0.00490806 −0.00320419	$0.03488062\times {10}^{-5}$ $0.03055420\times {10}^{-5}$ $0.02152411\times {10}^{-5}$ $0.12228008\times {10}^{-5}$ $0.10529440\times {10}^{-5}$ $0.07317913\times {10}^{-5}$ $0.45238704\times {10}^{-5}$ $0.29436702\times {10}^{-5}$ $0.15744064\times {10}^{-5}$ $0.13464271\times {10}^{-5}$ $0.56107579\times {10}^{-5}$ $0.82392769\times {10}^{-5}$ $0.49582843\times {10}^{-5}$
0.006	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.00017907 −0.00015740 −0.00011119 −0.00062919 −0.00054836 −0.00038476 −0.00249146 −0.00169177 −0.00081086 −0.00070427 −0.00315158 −0.00488287 −0.00318902	−0.00018011 −0.00015831 −0.00011183 −0.00063285 −0.00055151 −0.00038695 −0.00250501 −0.00170059 −0.00081557 −0.00070831 −0.00316838 −0.00490755 −0.00320388	$0.10442703\times {10}^{-5}$ $0.09147569\times {10}^{-5}$ $0.06444140\times {10}^{-5}$ $0.36609054\times {10}^{-5}$ $0.31525304\times {10}^{-5}$ $0.21910782\times {10}^{-5}$ $1.35475044\times {10}^{-5}$ $0.88167267\times {10}^{-5}$ $0.47135843\times {10}^{-5}$ $0.40312940\times {10}^{-5}$ $1.68036255\times {10}^{-5}$ $2.46805777\times {10}^{-5}$ $1.48565750\times {10}^{-5}$
0.010	(0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9)	−0.00017835 −0.00015677 −0.00011074 −0.00062669 −0.00054620 −0.00038326 −0.00248219 −0.00168574 −0.00080763 −0.00070152 −0.00314008 −0.00486597 −0.00317884	−0.00018009 −0.00015829 −0.00011182 −0.00063278 −0.00055144 −0.00038691 −0.00250473 −0.00170041 −0.00081547 −0.00070822 −0.00316804 −0.00490704 −0.00320357	$0.17368796\times {10}^{-5}$ $0.15214881\times {10}^{-5}$ $0.10718466\times {10}^{-5}$ $0.60890470\times {10}^{-5}$ $0.52437407\times {10}^{-5}$ $0.36446544\times {10}^{-5}$ $2.25390728\times {10}^{-5}$ $1.46707725\times {10}^{-5}$ $0.78399576\times {10}^{-5}$ $0.67055440\times {10}^{-5}$ $2.79583768\times {10}^{-5}$ $4.10722765\times {10}^{-5}$ $2.47304995\times {10}^{-5}$

,

Table 4. Comparison of absolute errors obtained for the solutions to $u\left(x, y, z, t\right)$ with absolute errors obtained by existing methods in the literature.

$\mathit{t}$	Present	LT-ADM [25]	LT-HPM [25]	VIM [25]	VIDM [25]	VIHPM [25]	CPU
0.002	$1.911\times {10}^{-8}$	$1.909\times {10}^{-8}$	$2.027\times {10}^{-8}$	$2.024\times {10}^{-8}$	$1.909\times {10}^{-8}$	$1.909\times {10}^{-8}$	47
0.004	$3.818\times {10}^{-8}$	$3.819\times {10}^{-8}$	$4.055\times {10}^{-8}$	$3.818\times {10}^{-8}$	$3.819\times {10}^{-8}$	$3.819\times {10}^{-8}$	78
0.006	$5.722\times {10}^{-8}$	$5.720\times {10}^{-8}$	$6.076\times {10}^{-8}$	$5.881\times {10}^{-8}$	$5.720\times {10}^{-8}$	$5.720\times {10}^{-8}$	105
0.008	$7.621\times {10}^{-8}$	$7.621\times {10}^{-8}$	$8.095\times {10}^{-8}$	$7.660\times {10}^{-8}$	$7.621\times {10}^{-8}$	$7.621\times {10}^{-8}$	156
0.010	$9.516\times {10}^{-8}$	$9.516\times {10}^{-8}$	$1.0108\times {10}^{-7}$	$9.552\times {10}^{-8}$	$9.516\times {10}^{-8}$	$9.516\times {10}^{-8}$	165

,

Table 5. Comparison of absolute errors obtained for the solutions to $v\left(x, y, z, t\right)$ with absolute errors obtained by existing methods in the literature.

$\mathit{t}$	Present	LT-ADM [25]	LT-HPM [25]	VIM [25]	VIDM [25]	VIHPM [25]
0.002	$1.0485948\times {10}^{-7}$	$1.0548528\times {10}^{-7}$	$1.1201619\times {10}^{-7}$	$1.0548547\times {10}^{-7}$	$1.0548528\times {10}^{-7}$	$1.0548463\times {10}^{-7}$
0.004	$2.0675305\times {10}^{-7}$	$2.1075463\times {10}^{-7}$	$2.2381514\times {10}^{-7}$	$2.1074306\times {10}^{-7}$	$2.1075463\times {10}^{-7}$	$2.1075204\times {10}^{-7}$
0.006	$3.0452328\times {10}^{-7}$	$3.1580841\times {10}^{-7}$	$3.3539724\times {10}^{-7}$	$3.1579848\times {10}^{-7}$	$3.1580841\times {10}^{-7}$	$3.1580257\times {10}^{-7}$
0.008	$3.9785722\times {10}^{-7}$	$4.2064707\times {10}^{-7}$	$4.4676292\times {10}^{-7}$	$4.2063130\times {10}^{-7}$	$4.2064707\times {10}^{-7}$	$4.2063669\times {10}^{-7}$
0.010	$4.8683879\times {10}^{-7}$	$5.2527104\times {10}^{-7}$	$5.5791260\times {10}^{-7}$	$5.2525095\times {10}^{-7}$	$5.2527104\times {10}^{-7}$	$5.2525483\times {10}^{-7}$

and

Table 6. Comparison of absolute errors obtained for the solutions to $w\left(x, y, z, t\right)$ with absolute errors obtained by existing methods in the literature.

$\mathit{t}$	Present	LT-ADM [25]	LT-HPM [25]	VIM [25]	VIDM [25]	VIHPM [25]
0.002	$7.0287830\times {10}^{-8}$	$7.031181\times {10}^{-8}$	$7.466501\times {10}^{-8}$	$7.028379\times {10}^{-8}$	$7.031181\times {10}^{-7}$	$7.031137\times {10}^{-8}$
0.004	$1.4021278\times {10}^{-7}$	$1.4047970\times {10}^{-7}$	$1.4918521\times {10}^{-7}$	$1.4047225\times {10}^{-7}$	$1.4047970\times {10}^{-7}$	$1.4047793\times {10}^{-7}$
0.006	$2.0945590\times {10}^{-7}$	$2.1050389\times {10}^{-7}$	$2.2356083\times {10}^{-7}$	$2.1048268\times {10}^{-7}$	$2.1050389\times {10}^{-7}$	$2.1049991\times {10}^{-7}$
0.008	$2.7772177\times {10}^{-7}$	$2.8038469\times {10}^{-7}$	$2.9779220\times {10}^{-7}$	$2.8041128\times {10}^{-7}$	$2.8038469\times {10}^{-7}$	$2.8037761\times {10}^{-7}$
0.010	$3.4478815\times {10}^{-7}$	$3.5012242\times {10}^{-7}$	$3.7187959\times {10}^{-7}$	$3.5009434\times {10}^{-7}$	$3.5012242\times {10}^{-7}$	$3.5011136\times {10}^{-7}$

.

Table 1, Table 2 and Table 3 present a comparison of exact solutions and numerical solutions obtained using the present method for $k=0.0001$ and different spatial points $\left(x, y, z\right)$ at different time instances ($t=0.002, 0.006, 0.010$). Additionally, they include the absolute errors between the two. It can be seen from the tables that the numerical solutions obtained using the present method closely match the exact solutions, as indicated by the small absolute error values. The largest errors appear at boundary points (e.g., $x=0.9$, $y=0.9$, $z=0.9$), indicating that the method might be slightly less accurate near domain edges. The central point $(0.5, 0.5, 0.5)$ generally shows smaller errors, suggesting that the numerical method is most stable at symmetric locations. The absolute errors are in the order of ${10}^{-5}$ or smaller, demonstrating the high accuracy of the method. The absolute errors tend to increase slightly as time progresses (from $t=0.002$ to $t=0.010$), but they remain relatively small. This suggests that the present method maintains good stability and accuracy over time.

Table 4, Table 5 and Table 6 show the comparison of absolute errors obtained by the present method with absolute errors obtained by the Laplace transform–Adomian decomposition method (LT-ADM) [25], Laplace transform homotopy perturbation method (LT-HPM) [25], variational iteration method (VIM) [25], variational iteration decomposition method (VIDM) [25], and variational iteration homotopy perturbation method (VIHPM) [25] at different times for grid point $\left(0.1,0.02,0.03\right)$.

Table 4 compares the absolute errors obtained using different numerical methods to solve the longitudinal velocity component $u\left(x,y,z,t\right)$. Absolute errors increase as time $t$ increases, which is expected in numerical approximations. The present method and LT-ADM [25] show almost identical errors at all time steps. LT-HPM [25] has the highest error values among the methods compared. VIM [23], VIDM [23], and VIHPM [23] show very close results to LT-ADM [25] and the present method, but small differences are observed. Overall, the errors remain in the order of ${10}^{-8}$, demonstrating high accuracy. The present method is one of the most accurate, with errors almost identical to LT-ADM [25]. LT-HPM [25] performs worse than other methods, indicating that it may introduce larger numerical errors.

Table 5 examines errors for the depth velocity component $v\left(x, y, z, t\right)$. Absolute errors increase with time in all methods. The error values are slightly higher than those in Table 4, indicating $v\left(x, y, z, t\right)$ has a marginally larger deviation. LT-ADM [25], VIDM [25], and VIHPM [25] show almost identical errors, meaning their numerical accuracy is very close. LT-HPM [25] has the highest errors, as seen in the previous table. The present method outperforms LT-ADM [25] at longer times, suggesting improved numerical accuracy. LT-HPM [25] has the highest error, making it the least reliable for depth velocity calculations.

Table 6 focuses on the transverse velocity component $w\left(x, y, z, t\right)$. As expected, absolute errors increase with increasing time. LT-ADM [25] and VIHPM [25] have the closest results to the present method. LT-HPM [25] continues to show the most errors, as seen in previous tables. The present method has the least errors overall, performing slightly better than LT-ADM [25]. LT-HPM [25] consistently underperforms, making it the least accurate method.

Table 7. $L_2$ and $L_{\infty }$ error norms for $\mathrm{Re}=100$, $h_x=h_y=h_z=h=0.05$, and $k=0.0001$ at different times.

$\mathit{t}$		${\mathit{L}}_2$			${\mathit{L}}_{\mathit{\infty }}$		CPU
	$\mathit{u}$	$\mathit{v}$	$\mathit{w}$	$\mathit{u}$	$\mathit{v}$	$\mathit{w}$
0.1	0.01607602	0.01701311	0.01701314	0.00078824	0.00052727	0.00052727	18
0.2	0.02885929	0.03118231	0.03118236	0.00136220	0.00094535	0.00094535	36
0.3	0.03925681	0.04314278	0.04314281	0.00180512	0.00129970	0.00129970	54
0.4	0.04777030	0.05327088	0.05327092	0.00211052	0.00158540	0.00158540	72
0.5	0.05473662	0.06183566	0.06183568	0.00234148	0.00182888	0.00182888	89
0.6	0.06040639	0.06904788	0.06904791	0.00256857	0.00203571	0.00203571	107
0.7	0.06497688	0.07508099	0.07508107	0.00273030	0.00220845	0.00220846	125
0.8	0.06860875	0.08008187	0.08008195	0.00283681	0.00234598	0.00234599	143
0.9	0.07144140	0.08418134	0.08418139	0.00289775	0.00245845	0.00245846	161
1.0	0.07357237	0.08747612	0.08747618	0.00292069	0.00254495	0.00254495	178

shows $L_2$ and $L_{\infty }$ error norms for $\mathrm{Re}=100$, $h_x=h_y=h_z=h=0.05$, and $k=0.0001$ at different times. It is observed from Table 7 that both $L_2$ and $L_{\infty }$ norms for velocity components $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ increase as time progresses, indicating a gradual accumulation in errors. The CPU time required for computation also increases with time, starting at 18 s at $t=0.1$ and reaching 178 s at $t=1$.

Table 8. $L_2$ and $L_{\infty }$ error norms at $t=1$ for $h_x=h_y=h_z=h=0.05$, $k=0.01$, and different kinematic viscosities.

$\mathbf{Re}$	${\mathit{L}}_2$			${\mathit{L}}_{\mathit{\infty }}$			CPU
	$\mathit{u}$	$\mathit{v}$	$\mathit{w}$	$\mathit{u}$	$\mathit{v}$	$\mathit{w}$
10	0.12595460	0.17624530	0.17628540	0.00398876	0.00496712	0.00496995
20	0.14303290	0.19327000	0.19330240	0.00473712	0.00550840	0.00551130
30	0.13336220	0.17461560	0.17464180	0.00461626	0.00496580	0.00496805
40	0.12094910	0.15458880	0.15460980	0.00431160	0.00443582	0.00443767
50	0.10965810	0.13755450	0.13757130	0.00401893	0.00393986	0.00394031	2
60	0.09998650	0.12355520	0.12356920	0.00377040	0.00356142	0.00356287
70	0.09178136	0.11202780	0.11203940	0.00352644	0.00324337	0.00324450
80	0.08479263	0.10243280	0.10244300	0.00329654	0.00296713	0.00296735
90	0.07879200	0.09434658	0.09435534	0.00308419	0.00273646	0.00273663
100	0.07359336	0.08744954	0.08745717	0.00291878	0.00254064	0.00254077
1000	0.01124273	0.01200673	0.01200697	0.00054696	0.00036272	0.00036273
10000	0.00122298	0.00127331	0.00127331	0.00006759	0.00003947	0.00003947	3

shows $L_2$ and $L_{\infty }$ error norms at $t=1$ for $h_x=h_y=h_z=h=0.05$, $k=0.01$, and different values of Re. It is observed from Table 7 that as Re increases, both $L_2$ and $L_{\infty }$ norms decrease, indicating improved accuracy. At very high Re values (e.g., 1000 and 10,000), the errors become significantly smaller, suggesting the method remains stable even for high Reynolds numbers. The error norms for $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ follow a similar decreasing trend with increasing Re.

Figure 2, Figure 3 and Figure 4 show the numerical solutions and exact solutions to $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ for $\mathrm{Re}=10$, $h_x=h_y=h_z=h=0.05$, and $k=0.0001$ with $z=0.5$ at $t=0.25$. It can be seen from Figure 2, Figure 3 and Figure 4 that both solutions align closely, indicating that the numerical method accurately approximates the exact solution. Also, the smoothness of the curves suggests numerical stability and precision.

Figure 5, Figure 6 and Figure 7 show numerical solutions and exact solutions to $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ for $\mathrm{Re}=100$, $h_x=h_y=h_z=h=0.05$, and $k=0.0001$ with $z=0.5$ at $t=0.02$. Each figure presents contour plots illustrating the spatial distribution of the solutions. The numerical and exact results exhibit strong agreement, confirming the accuracy of the proposed numerical method.

Figure 8, Figure 9 and Figure 10 show a comparison between the numerical solutions and exact solutions of $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ for $\mathrm{Re}=10$, $h_x=h_y=h_z=h=0.05$, and $k=0.0001$ with $x=0.5$ and $z=0.5$ at $t=0.05$. These comparisons are essential for validating the numerical method. Consistency across all components suggests that the scheme handles 3D flow accurately at this Reynolds number and time step.

Problem 2: We consider the three-dimensional coupled Burgers’ Equations (1)–(3) with initial conditions:

$$\begin{array}{l}u\left(x,y,z,0\right)=\sin \left(\pi x\right)\sin \left(\pi y\right)\sin \left(\pi z\right), \\ v\left(x,y,z,0\right)=\left(\sin \left(\pi x\right)+\sin \left(2\pi x\right)\right)\left(\sin \left(\pi y\right)+\sin \left(2\pi y\right)\right)\left(\sin \left(\pi z\right)+\sin \left(2\pi z\right)\right), \\ w\left(x,y,z,0\right)=\left(\sin \left(\pi x\right)+\sin \left(4\pi x\right)\right)\left(\sin \left(\pi y\right)+\sin \left(4\pi y\right)\right)\left(\sin \left(\pi z\right)+\sin \left(4\pi z\right)\right). \end{array}$$

and boundary conditions:

$$u\left(0,y,z,t\right)=u\left(1,y,z,t\right)=u\left(x,0,z,t\right)=u\left(x,1,z,t\right)=u\left(x,y,0,t\right)=u\left(x,y,1,t\right)=0,$$

$$v\left(0,y,z,t\right)=v\left(1,y,z,t\right)=v\left(x,0,z,t\right)=v\left(x,1,z,t\right)=v\left(x,y,0,t\right)=v\left(x,y,1,t\right)=0,$$

$$w\left(0,y,z,t\right)=w\left(1,y,z,t\right)=w\left(x,0,z,t\right)=w\left(x,1,z,t\right)=w\left(x,y,0,t\right)=w\left(x,y,1,t\right)=0.$$

Since the exact solution to this problem is not available, numerical solutions were obtained for the values of $\mathrm{Re}$, $k$, $h_x=h_y=h_z=h$, and $t$ from the literature. The results were then compared with those available in the literature. In all numerical computations, we used the mesh width $h_x=h_y=h_z=h=0.05$ and $k=0.0001$ for problem 2.

Table 9. Numerical solutions for $\mathrm{Re}=20$ at different times.

Grid Point		$\mathit{t}=0.05$	$\mathit{t}=0.1$	$\mathit{t}=0.25$	$\mathit{t}=0.5$	$\mathit{t}=1$
(0.1, 0.1, 0.9)	$u$ $v$ $w$	0.0239584 −0.1433502 −0.3098060	0.0251936 −0.0942885 −0.0444397	0.0198349 −0.0019673 0.0571594	0.0087784 0.0161314 0.0209709	0.0025408 0.0079498 0.0053030
(0.1, 0.1, 0.1)	$u$ $v$ $w$	0.0130397 0.2961703 0.5933083	0.0086755 0.1819000 0.2619675	0.0042517 0.0713250 0.0521275	0.0021648 0.0258374 0.0125001	0.0010801 0.0071994 0.0034688
(0.9, 0.1, 0.1)	$u$ $v$ $w$	0.0373669 −0.2133582 −0.3512228	0.0350676 −0.1539166 −0.0849666	0.0219688 −0.0399523 0.0298473	0.0091082 0.0024782 0.0171387	0.0024632 0.0052484 0.0049001
(0.9, 0.1, 0.9)	$u$ $v$ $w$	0.0287756 0.0465253 0.1132805	0.0261095 0.0281507 0.0222440	0.0290998 0.0072191 0.0443088	0.0224096 0.0118066 0.0329812	0.0056462 0.0084688 0.0085465
(0.5, 0.5, 0.5)	$u$ $v$ $w$	0.8804632 1.7752066 0.2335210	0.6788833 2.0130337 0.1981508	0.3317227 1.3315012 0.3736063	0.1704326 0.6964189 0.3278693	0.0806389 0.2940547 0.1650422
(0.1, 0.9, 0.9)	$u$ $v$ $w$	0.0293427 0.0473384 0.1131651	0.0258647 0.0287051 0.0199498	0.0209887 0.0078080 0.0268328	0.0157792 0.0255877 0.0232304	0.0063097 0.0232738 0.0108606
(0.1, 0.9, 0.1)	$u$ $v$ $w$	0.0285317 −0.1743388 −0.3571369	0.0222670 −0.1143676 −0.1186861	0.0122100 −0.0169228 0.0035327	0.0080087 0.0576209 0.0156805	0.0027520 0.0189870 0.0066534
(0.9, 0.9, 0.1)	$u$ $v$ $w$	0.0274635 0.0434720 0.1061607	0.0280131 0.0281587 0.0114414	0.0204932 0.0042559 0.0088801	0.0117795 0.0117706 0.0139944	0.0049298 0.0145751 0.0080140
(0.9, 0.9, 0.9)	$u$ $v$ $w$	0.0271956 −0.0122070 −0.0415867	0.0264742 −0.0067082 −0.0019528	0.0306129 0.0000573 0.0280496	0.0370216 0.0201885 0.0365225	0.0153747 0.0316911 0.0195732
CPU		4	8	19	38	75

presents numerical solutions for different grid points at various time steps for $\mathrm{Re}=20$. At $t=0.5$, the velocity components $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ are at their highest values. As time progresses to $t=1$, the values decrease significantly across all grid points, indicating a decay in velocity magnitudes due to viscous dissipation. The central point $\left(0.5, 0.5, 0.5\right)$ has the highest velocity values for $u\left(x, y, z, t\right)$ and $v\left(x, y, z, t\right)$, suggesting a peak in velocity at the core of the computational domain. Edge points like $\left(0.9, 0.9, 0.9\right)$ and $\left(0.1, 0.1, 0.1\right)$ exhibit lower initial velocities, which could be due to boundary effects or flow stabilization over time. The numerical scheme maintains smooth transitions in velocity over time. The decay in velocity values confirms that the numerical method handles dissipation effects well. The highest velocity magnitudes are seen at the central point and decrease outward toward the boundaries. The present method ensures a stable solution without sudden numerical fluctuations.

Table 10. Comparison of numerical solutions obtained by the present method with the other numerical solutions obtained by existing methods for $\mathrm{Re}=10$ at different times.

Grid Point		Method	$\mathit{t}=0.05$	$\mathit{t}=0.25$	$\mathit{t}=0.5$	$\mathit{t}=1$
(0.1, 0.1, 0.9)	$u$	Present DQ-FDM [24] FDM [24]	0.0254111 0.02364 0.025403	0.0115768 0.01076 0.011756	0.0041839 0.00391 0.004266	0.0009462 0.00090 0.000983
	$v$	Present DQ-FDM [24] FDM [24]	−0.1071553 −0.10201 −0.108493	0.0109503 0.00971 0.010697	0.0093759 0.00872 0.009521	0.0024052 0.00227 0.002498
	$w$	Present DQ-FDM [24] FDM [24]	−0.0680342 −0.07772 −0.089963	0.0185890 0.01747 0.020117	0.0058328 0.00545 0.005872	0.0012179 0.00115 0.001228
(0.1, 0.1, 0.1)	$u$	Present DQ-FDM [24] FDM [24]	0.0135700 0.01291 0.013941	0.0043702 0.00413 0.004521	0.0022730 0.00214 0.002331	0.0007671 0.00073 0.000798
	$v$	Present DQ-FDM [24] FDM [24]	0.2722870 0.26139 0.284202	0.0456060 0.04368 0.048848	0.0124520 0.01180 0.013223	0.0023299 0.00221 0.002449
	$w$	Present DQ-FDM [24] FDM [24]	0.3243355 0.32746 0.379733	0.0148860 0.01391 0.016177	0.0042409 0.00391 0.004263	0.0010379 0.00097 0.001045
(0.5, 0.5, 0.5)	$u$	Present DQ-FDM [24] FDM [24]	0.8221614 0.82203 0.826809	0.3196220 0.32429 0.328282	0.1489025 0.15109 0.152905	0.0361105 0.03715 0.037998
	$v$	Present DQ-FDM [24] FDM [24]	1.5623827 1.57094 1.540632	0.9474787 0.94780 0.971748	0.4004099 0.40393 0.409879	0.0912639 0.09344 0.094790
	$w$	Present DQ-FDM [24] FDM [24]	0.3610981 0.37251 0.401473	0.3579388 0.35500 0.340496	0.1880154 0.18819 0.185145	0.0451686 0.04595 0.045831
(0.9, 0.9, 0.1)	$u$	Present DQ-FDM [24] FDM [24]	0.0266505 0.02445 0.026479	0.0129122 0.01202 0.013229	0.0063366 0.00592 0.006476	0.0012449 0.00118 0.001334
	$v$	Present DQ-FDM [24] FDM [24]	0.0279643 0.02662 0.029084	0.0038226 0.00378 0.003146	0.0113319 0.01041 0.010327	0.0030525 0.00288 0.003158
	$w$	Present DQ-FDM [24] FDM [24]	0.0139741 0.02232 0.032489	0.0133409 0.01219 0.012178	0.0068297 0.00626 0.006790	0.0015038 0.00141 0.001544
(0.9, 0.9, 0.9)	$u$	Present DQ-FDM [24] FDM [24]	0.0259294 0.02393 0.025940	0.0206922 0.01909 0.020358	0.0103278 0.00958 0.010436	0.0015613 0.00149 0.001669
	$v$	Present DQ-FDM [24] FDM [24]	−0.0067402 −0.00660 −0.007179	0.0023861 0.00250 0.002243	0.0104624 0.00969 0.009546	0.0032999 0.00311 0.003387
	$w$	Present DQ-FDM [24] FDM [24]	−0.0027490 −0.00627 −0.006416	0.0218266 0.02055 0.020134	0.0103696 0.00968 0.010395	0.0018227 0.00172 0.001875
CPU			3	19	38	75

and

Table 11. Comparison of numerical solutions obtained by the present method with the other numerical solutions obtained by existing methods for $\mathrm{Re}=4$ at different times.

Grid Point		Method	$\mathit{t}=0.05$	$\mathit{t}=0.25$	$\mathit{t}=0.5$	$\mathit{t}=1$
(0.1, 0.1, 0.9)	$u$	Present DQ-FDM [24] FDM [24]	0.0207517 0.01919 0.020817	0.0038962 0.00370 0.004033	0.0006396 0.00062 0.000682	0.0000164 0.00002 0.000019
	$v$	Present DQ-FDM [24] FDM [24]	−0.0310491 −0.03350 −0.035715	0.0066557 0.00622 0.006890	0.0010418 0.00100 0.001105	0.0000260 0.00003 0.000030
	$w$	Present DQ-FDM [24] FDM [24]	0.0279222 0.02000 0.026177	0.0042174 0.00401 0.004361	0.0006661 0.00064 0.000706	0.0000170 0.00002 0.000020
(0.1, 0.1, 0.1)	$u$	Present DQ-FDM [24] FDM [24]	0.0131351 0.01247 0.013573	0.0031357 0.00297 0.003242	0.0006058 0.00058 0.000645	0.0000164 0.00002 0.000019
	$v$	Present DQ-FDM [24] FDM [24]	0.1831330 0.17987 0.197984	0.0099416 0.00978 0.010907	0.0010810 0.00105 0.001159	0.0000260 0.00003 0.000030
	$w$	Present DQ-FDM [24] FDM [24]	0.0690051 0.07888 0.095820	0.0035683 0.00339 0.003684	0.0006332 0.00061 0.000746	0.0000169 0.00002 0.000020
(0.5, 0.5, 0.5)	$u$	Present DQ-FDM [24] FDM [24]	0.6654563 0.66955 0.672757	0.1411316 0.14484 0.146526	0.0223667 0.02345 0.023959	0.0005572 0.00063 0.000654
	$v$	Present DQ-FDM [24] FDM [24]	1.0757338 1.06574 1.055410	0.2289045 0.23342 0.233923	0.0354140 0.03684 0.037139	0.0008814 0.00098 0.001013
	$w$	Present DQ-FDM [24] FDM [24]	0.5259060 0.53003 0.531746	0.1456256 0.14901 0.149800	0.0231289 0.02420 0.024597	0.0005761 0.00065 0.000672
(0.9, 0.9, 0.1)	$u$	Present DQ-FDM [24] FDM [24]	0.0205042 0.01899 0.020760	0.0045598 0.00431 0.004726	0.0006817 0.00066 0.000733	0.0000165 0.00002 0.000019
	$v$	Present DQ-FDM [24] FDM [24]	0.0047205 0.00592 0.006935	0.0051581 0.00468 0.004731	0.0010428 0.00100 0.001076	0.0000260 0.00003 0.000030
	$w$	Present DQ-FDM [24] FDM [24]	0.0080648 0.00749 0.0053180	0.0045400 0.00428 0.004689	0.0006999 0.00068 0.000747	0.0000170 0.00002 0.000020
(0.9, 0.9, 0.9)	$u$	Present DQ-FDM [24] FDM [24]	0.0212308 0.01971 0.021345	0.0053239 0.00502 0.005498	0.0007187 0.00070 0.000775	0.0000165 0.00002 0.000019
	$v$	Present DQ-FDM [24] FDM [24]	−0.0006289 −0.00100 −0.001143	0.0035969 0.00323 0.003274	0.0010081 0.00096 0.001032	0.0000260 0.00003 0.000030
	$w$	Present DQ-FDM [24] FDM [24]	0.0051624 0.00529 0.004513	0.0053273 0.00503 0.005506	0.0007364 0.00071 0.000788	0.0000171 0.00002 0.000020
CPU			4	19	37	75

present the comparison of the computed values of $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ with those given by DQ-FDM [24] and FDM [24] for $\mathrm{Re}=10$ and $\mathrm{Re}=4$, respectively. The present method is consistently close to DQ-FDM [24] and FDM [24] at all time steps. For $u\left(x, y, z, t\right)$ velocity, the present method and DQ-FDM [24] show very small differences, usually within the order of ${10}^{-4}$. FDM [24] tends to have slightly higher deviations, particularly for the larger time value of $t=1$. For $v\left(x,y,z,t\right)$ velocity, the present method and DQ-FDM [24] are nearly identical, with maximum deviations around ${10}^{-3}$. FDM [24] sometimes deviates more, particularly for higher Reynolds numbers, indicating it may not capture velocity decay as efficiently. For $w\left(x,y,z,t\right)$ velocity, the present method generally outperforms both DQ-FDM [24] and FDM [24] in maintaining consistency across time. FDM [24] shows the highest deviations from the other two methods, suggesting that the finite difference approach introduces more numerical diffusion. The present method maintains stability better than FDM [24] at larger time steps (especially at $t=1$). DQ-FDM [24] is also highly stable, showing results very close to the present method, making both methods strong candidates for solving such equations. FDM [24] sometimes exhibits noticeable differences at later time steps, indicating a slight loss of accuracy over time. Consequently, the present method provides the best balance between accuracy and efficiency. DQ-FDM [24] is comparable in accuracy but requires higher computational effort. FDM [24] shows the largest deviations, particularly for longer time steps, making it less suitable for high-accuracy simulations. Overall, the present method is a strong and reliable numerical approach to solving three-dimensional Burgers’ equations.

Figure 11, Figure 12 and Figure 13 show $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ numerical solutions for $\mathrm{Re}=10$ at $t=0.25$. These contour plots provide a visual representation of the numerical results, confirming the expected flow behavior. Additionally, the solution graphs for $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ at $z=0.5$ suggest partial symmetry along the $y$-axis.

Figure 14, Figure 15 and Figure 16 show $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ numerical solutions for $x=0.5$, $z=0.5$, and $\mathrm{Re}=4$ at $t=0.1, 0.2, 0.3$. Although small deviations from perfect symmetry are observed, the $u\left(x, y, z, t\right)$, $v\left(x, y, z, t\right)$, and $w\left(x, y, z, t\right)$ graphs exhibit a certain degree of symmetry along the $y$-axis.

4. Conclusions

In this study, to obtain the numerical solutions to three-dimensional coupled Burgers’ equations, we first applied the fractional step method to split the equations into one-dimensional equations, and then, we used the implicit finite difference approximations to solve the obtained one-dimensional equations. Two test problems were used to illustrate the accuracy of the present schemes. Comparisons were made with the existing methods in the literature. The method was analyzed by the von Neumann stability analysis method, and it was demonstrated that the method is unconditionally stable. Overall, the results confirm that the proposed method is a highly effective and reliable numerical tool for solving three-dimensional coupled Burgers’ equations and can be extended to other complex three-dimensional systems.

5. Note

The computations were performed using Fortran PowerStation version 4.0 software and graphically depicted using EXCEL 2016 software.