A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation

Abstract

A numerical solution of the one-dimensional Burgers’ equation is obtained using a sixth-order compact finite difference method. To achieve this, a tridiagonal sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge–Kutta scheme in time have been combined. The scheme is implemented to solve two test problems with known exact solutions. Comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort. The present results are also seen to be more accurate than some available results given in the literature.

Introduction

Burgers’ equation has attracted much attention in studying evolution equations describing wave propagation, investigating the shallow water waves [1], [2] and in examining the chemical reaction–diffusion model of Brusselator [3]. Not only has Burgers’ equation been found to describe various phenomena such as a mathematical model of turbulence [4], it is also a very important fluid dynamical model both for the conceptual understanding of physical flow and testing various new solution approaches. Moreover, simulation of Burgers’ equation is a natural first step towards developing methods for computations of complex flows. The existence and uniqueness of classical solutions to the generalized Burgers equation have been proved with certain conditions [5], [6].

In recent years, computing the solution of Burgers’ equation has attracted a lot of attention. As exact solutions in terms of infinite series fail for small values of viscosity [7], $v<0.01$, many authors [8], [9], [10], [11], [12], [13] have used various numerical techniques based on finite difference, finite element, cubic spline function, pseudo-spectral and boundary elements in attempting to solve the equation.

Throughout the last two decades, second-order numerical schemes were considered to be sufficient for most flow problems. In particular, the central and upwind schemes have proved the most popular because of their ease of application in applied fields of science. Although most problems often give quite good results on reasonable meshes, the solution may be of poor quality for convection dominated flows if the mesh is not sufficiently refined. In the meantime, higher order discretization is generally associated with non-compact stencils which increase the bandwidth of the resultant coefficient matrix. Both mesh refinement and increased matrix bandwidth always lead to a large number of arithmetic operations. Thus, neither lower order accurate schemes on a fine mesh nor higher order accurate schemes on a non-compact stencil seem to be computationally cost-effective. This is why compact finite difference methods are important. The higher order accuracy combined with the compactness of the difference stencils gives highly accurate numerical solutions on relatively coarser grids with greater computational efficiency [14]. Various versions of the compact schemes have been analyzed and implemented successfully by some researchers [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].

Hassanien et al. [29] proposed a two-level three-point finite difference method of order 2 in time and 4 in space by computing the local truncation error. They solved the problem very successfully with the use of a second-order implicit time integration scheme.

To compute the solutions of the problem, the present work attempts to combine a low-storage third-order total variation diminishing Runge–Kutta (TVD-RK3) scheme [30] in time and a sixth-order compact finite difference (CFD6) scheme in space. The combined scheme needs relatively less storage space and is more accurate than the literature [9], [11], [29], [31], [32], [33], [34]. Unlike some previous works, there is no need linearization of the nonlinear terms.

Here a numerical solution of the one-dimensional Burgers’ equation is obtained using the CFD6 method. To the best of the authors’ knowledge, this combination has not previously been implemented for the Burgers’ equation. Numerical computations show that the present method offers better accuracy in comparison with previous work. Furthermore, the current method here is more general and can therefore be used for solving nonlinear partial differential equations arising in various areas.

Behaviour of many physical systems encountered in models of traffic and fluid flow leads to Burgers’ equation. The following one-dimensional Burgers’ equation problem arising in various fields of science is considered,

$$u_t+{\mathit{uu}}_x-\upsilon u_{\mathit{xx}}=0\text{,}a<x<b\text{,}t>0\text{,}$$

with the boundary conditions

$$\begin{array}{cc}\hfill & u(a\text{,}t)=f_1(t)\text{,}t>0\text{,}\hfill \\ \hfill & u(b\text{,}t)=f_2(t)\text{,}t>0\text{,}\hfill \end{array}$$

and the initial condition

$$u(x\text{,}0)=g(x)\text{,}a<x<b\text{,}$$

where $u(x\text{,}t)$ indicates the velocity for space x and time t, $\upsilon >0$ is the kinematic viscosity parameter, $f_1\text{,}{f}_2$ and g are known functions of their arguments, $a>0$ and $b>0$ are real constants. In Eqs. (1), (2), (3), x and t denote derivatives with respect to space and time, respectively, when they are used as subscripts.

This paper proposes a numerical scheme to solve the considered Burgers’ Eq. (1) with a set of boundary and initial conditions given by Eqs. (2), (3). The current work aims to demonstrate that the scheme is capable of achieving high accuracy for the problem under consideration. The organization of this paper is as follows. The scheme is summarized and applied to Burgers’ equation in Section 2. The accuracy and efficiency of the method is investigated with some numerical illustrations in Section 3. Section 4 consists of some concluding remarks.