A parameter-uniform implicit difference scheme for solving time-dependent Burgers’ equations

摘要

A numerical study is made for solving one dimensional time dependent Burgers’ equation with small coefficient of viscosity. Burgers’ equation is one of the fundamental model equations in the fluid dynamics to describe the shock waves and traffic flows. For high coefficient of viscosity a number of solution methodology exist in the literature [6], [7], [8], [9] and [14] but for the sufficiently low coefficient of viscosity, the exist solution methodology fail and a discrepancy occurs in the literature. In this paper, we present a numerical method based on finite difference which works nicely for both the cases, i.e., low as well as high viscosity coefficient. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on uniform mesh and a standard upwind finite difference scheme to discretize in spacial direction on piecewise uniform mesh. The quasilinearzation process is used to tackle the non-linearity. An extensive amount of analysis has been carried out to obtain the parameter uniform error estimates which show that the resulting method is uniformly convergent with respect to the parameter. To illustrate the method, numerical examples are solved using the presented method and compare with exact solution for high value of coefficient of viscosity.