In this paper we use the Arakawa Jacobian method [1] and the fourth-order essentially non-oscillatory (ENO-4) scheme of Osher and Shu [15] to solve the equatorial beta-plane barotropic equations. The Arakawa Jacobian scheme is a second order centred finite differences scheme that conserves energy and enstrophy. The fourth-order essentially non-oscillatory scheme is designed for Hamilton-Jacobi equations and traditionally used to track sharp fronts. We are interested in the performance of these two methods on the baratropic equations and determine whether they are adequate for studying the barotropic instability. The two methods are tested and compared on two typical exact solutions, a smooth Rossby wave-packet and a discontinuous shear, on the long-climate scale of 100 days. The numerical results indicate that the Arakawa Jacobian method conserves energy and enstrophy nearly exactly, as expected, captures the phase speed the Rossby wave, and achieves an overall second order accuracy, in both cases. The same properties are preserved by the ENO-4 scheme but the fourth order accuracy is observed only for the smooth Rossby wave solution while in the case of the discontinuous shear, it yields an overall third order accuracy, even in the smooth regions, away from the discontinuity.