## 1. Introduction

Burgers’ equation with

${\nu}_{d}$ as coefficient of viscosity can be defined as

where

with boundary conditions (BCs),

and initial condition (IC),

Linearized form of Burgers’ equation (by using Hopf-Cole transformation) is given as

with the Neumann boundary conditions (BCs),

and the initial condition (IC),

Burgers’ equation is a very simple form of the Navier–Stokes equation and it always attracted researchers due to its occurrence in several areas of physics and applied mathematics, like fluid mechanics, gas dynamics, traffic flow, in the theory of shock waves, and nonlinear acoustics. Firstly, it originated from Bateman [

1] in 1915. Later, in 1948, JM Burgers studied it as a class of equation [

2,

3] to mathematically delineate the turbulence model. Recently, in 2019, Ryu et al. [

4] proposed some nowcasting rainfall models that are based on Burger’s equation. Existence and uniqueness of the solution of Equation (

1) and its generalized form can be found in [

5,

6,

7].

Recently, due to the availability of high-speed computers, activities that are related to the computation of numerical solution has increased. Özis et al. [

8] used finite element approach to solve Burgers’ equation. Dogan [

9] proposed a Galerkin finite element method to solve Burgers’ equation. A quadratic and cubic spline collocation method was developed in the paper [

10,

11,

12,

13]. Elgindy et al. [

14] developed a higher-order numerical scheme by using Hopf-Cole barycentric Gegenbauer integral pseudospectral method. Korkmaz et al. [

15] and Jiwari et al. [

16] established polynomial based and weighted average based differential quadrature scheme, respectively to solve Burgers’ equation. Verma et al. [

17,

18] developed Du Fort-Frankel and Douglas finite difference scheme that are unconditionally stable to solve Burgers’ equation. Hassanien et al. [

19] developed a two-level three-point finite difference scheme of order 2 in time and order 4 in space to solve Burgers’ equation. Wavelet-based numerical schemes have been developed in [

20,

21]. Chebyshev collocation method was used in [

22,

23,

24] in order to solve Burgers’ equation, Volterra–Fredholm integral equation, and Riemann-Liouville and Riesz fractional advection-dispersion problems, respectively. Gowrisankar et al. [

25] studied singularly perturbed Burgers’ equation.

A well known scheme for diffusion equation is Crank–Nicolson (CN) [

26,

27,

28,

29]. CN is a second-order scheme based on Trapezoidal formula which is A-stable but not

L-stable. In the presence of inconsistencies [

30] or when time step taken is large [

27], CN produces unwanted oscillations. As an alternative to CN, Chawla et al. [

31] proposed generalized Trapezoidal formula (GTF(

$\alpha $)) with

$\alpha >0$, which is

L-stable and gives stable results. Chawla et al. [

32] proposed a modified Simpson’s

$1/3$ rule (ASIMP), which is A-stable, and used it to give fourth-order time integration formula for diffusion equation, but it suffers from producing unwanted oscillations like CN as it lacks L-stability. To remove these unwanted oscillations, Chawla et al. [

33], Lajja et al. [

34], and Verma et al. [

35] proposed and analyzed various types of

L-stable methods, which provides accurate and stable results.

The Burger’s Equation (

1) subject to some BCs and IC has an exact solution in the form of Fourier series, which does not converge for small values of viscosity. Hence, it always attracts researchers to test newly developed numerical methods on this nonlinear parabolic PDE.

Here, we derive 7th order time integration formula that is weakly

L-stable and generalize the results presented in [

33,

35]. The issue of slow convergence of series solution for small

${\nu}_{d}$ forces analytical solution of Equation (

1) to diverge from the true solution and, hence, for small values of

${\nu}_{d}$, it is not easy to compute the solutions. The newly developed method computes the solution even for small

${\nu}_{d}$. Additionally, it provides satisfactory results in the case of inconsistencies.

We discuss truncation error, stability in detail, in order to show that the developed scheme is convergent. We use software Mathematica 11.3 to compute the solution and Origin $8.5$ for the plotting purpose.

The remainder of this article is constructed, as follows. In

Section 2, we give close form solution, which we use to compute the exact solution. In

Section 3, we derive a higher-order time integration method for

${u}^{\prime}\left(t\right)=f(t,u)$. In

Section 4, we derive a numerical scheme for the Burgers’ equation. In

Section 5, we illustrate the numerical results with tables and two-dimensional (2D)–three-dimensional (3D) graphs.