Comparison of Godunov’s and relaxation schemes approximation of solutions to the traffic flow and Euler equations

Abstract

The study of Euler equations in gas dynamics is well known to give birth to the theory of hyperbolic conservation law, that is, Euler equations are essential in development, analysis and successful use of numerical methods for non-linear systems of conservation laws particularly in problems involving shock waves. This research deals with the study of Euler equations for isothermal gas as well as traffic flow model that is governed by two hyperbolic equations. The equations are analysed to obtain two real and distinct eigen values which enables one to determine the wave structure of the possible solution to the Riemann problem set up. The numerical solution to the Riemann problem set up is obtained using both the Godunov scheme and the relaxation scheme. Finally, comparison of the results obtained from these two schemes is done graphically and Relaxation scheme appears to be more promising and a good alternative scheme as compared to Godunov scheme because of its simplicity.