Burgers Equation

Abstract

In the previous chapter (chapter 6) we derived the Green element equations for the mathematical statement that describes the storage and movement of species or contaminants in a fluid. The transported specie can be considered to be the momentum of the flow, and in that case the relevant differential equations become the Navier-Stokes equations [1]. Simplifications of the Navier-Stokes equations can be carried out to achieve easier versions of the equations. One of such second-order simplifications yields the transient one-dimensional form of the equations which provides a useful model for many physical fluid flow phenomena — nonlinear propagating shock wave with viscous dissipation, turbulence, propagating shock waves in gases, propagating flame in a combustion chamber, etc. As a result of the extensive research works carried out by Burgers in modeling of turbulence, the simplified transient nonlinear momentum transport equation in one spatial dimension is popularly referred to as Burgers equation [2,3]. The nonlinear nature of Burgers equation has been exploited as a useful prototype differential equation for modeling many divers and rather unrelated phenomena such as shock flows, wave propagation in combustion chambers, vehicular traffic movement, acoustic transmission, etc. In fact, Burgers equation can be considered applicable to any flow phenomenon in which there exist the balancing effects of viscous and inertia or convective forces. It is probably one of the simplest nonlinear transient partial differential equations which exhibits some very unique features. When inertia or convective forces are dominant, its solution resembles that of the kinematic wave equation which displays a propagating wave front and boundary layers. In that case Burgers equation essentially behaves as a hyperbolic partial differential equation. In contrast, when viscous forces are dominant, it behaves as a parabolic equation, and any propagating wave front is smeared and diffused due to viscous action. Because of these different forms that Burgers equation can assume, coupled with its nonlinear characteristics, it has become a model equation for assessing and evaluating the performance of many computational techniques. It is for the same reason that we have devoted this chapter to the development of a number of schemes of the Green element method for the solution of the Burgers equation.