The main objective of this research work is to apply the discontinuous Galerkin method to a classical partial differential equation to investigate the properties of the numerical solution and compare the numerical solution to the analytical solution by using discontinuous Galerkin method. This scheme is applied to 1-D non-linear conservation equation (Burgers equation) in which the governing differential equation is simplified model of the inviscid Navier-stokes equations. In this work three cases are studied. They are sinusoidal wave profile, initial shock discontinuity and initial linear distribution. A grid and time step refinement is performed. Riemann fluxes at each element interfaces are calculated. This scheme is applied to forward differentiation method (Euler's method) and to second order Runge-kutta method of this work.