## 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.

## Keywords

Burger Equation Green Element Element Equation Hyperbolic Partial Differential Equation Propagate Shock Wave## Preview

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## References

- 1.Schlichting, H.,
*Boundary-Layer Theory*, McGraw-Hill Book Comp., 7th Ed., 1979.Google Scholar - 2.Burgers, J.M., “A mathematical Model illustrating the theory of Turbulence”,
*Advances in Applied Mech*., (1), Ed. by R. von Mises and T. von Karman, Academy Press, NY., pp. 171–199, 1948.Google Scholar - 3.Hopf, E., “The partial differential equation
*u t*+*u u X*=*µu m ” , Comm. Pure Appl. Math*., 3, pp. 201–230, 1950.Google Scholar - 4.Taigbenu, A.E., “The Green Element Method,”
*Int. J. for Numerical Methods in Engineering*, Vol. 38, pp 2241–2263, 1995.CrossRefGoogle Scholar - 5.Benton, E.R. and G.W. Platzman, “A table of solutions of the one-dimensional Burgers equation”,
*Quarterly of Applied Mathematics*, pp. 195–212, July, 1972.Google Scholar - 6.Taigbenu„ A.E. and O.O. Onyejekwe, “A Mixed Green element formulation for transient Burgers’ equation”,
*Int. J. Num. Methods in Fluids*,**24**, 563–578, 1997.CrossRefGoogle Scholar - 7.Lighthill, M.J., “Viscosity effects in sound waves of finite amplitude”, in
*Surveys in Mechanics*, Ed.: G.K. Batchelor and R.M. Davis, cambridge Univ. Press, 1956.Google Scholar - 8.Fletcher, C.A.J., “Burger’s Equation: A model for all reasons”,
*Numerical Sol. of Partial Differential Equations*, Ed. by J. Noye, North-Holland Pub. Comp. pp. 139–225, 1982.Google Scholar - Kakuda, K. and N. Tosaka, “The generalized boundary element approach to Burgers equation”,
*Mt. J. Num. Meth. in Engrg*.,29, pp. 245–261, 1990.Google Scholar - 10.Cole J.D., “On a quasi-linear parabolic equation occurring in aerodynamics”,
*Quarterly of Applied mathematics*, 23, pp. 225–236 1951.Google Scholar - 11.Varoglu, E. and W.D.L. Finn, Space-Time finite elements incorporating characteristics for the Burgers’ equations,
*Int. J. Num. Meth. in Engrg*.,16, 171–184, 1980.Google Scholar