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

Keywords: Compact schemes Finite difference method Burgers' equation Low-storage Runge–Kutta scheme a b s t r a c t 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. 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 [8–13] 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 dis-cretization 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 …