In order to fill the gap between the results of numerical simulations and experiments, several factors have to be considered and one among them is the accuracy and reliability of numerical scheme employed in the simulations. If we use the traditional second order accurate central difference method, they suffer from computational instability and may not converge when convective terms dominate. While the upwind method suppresses the unwanted physical oscillations and enables us to get solutions for a large range of cell Reynolds numbers, the major disadvantage associated with the upwind method is that its order of accuracy is very low, which is where is the grid size. In the past, in order to get optimal solution for the wide range of parameters, researchers generate benchmark results by applying the central difference operator to diffusion terms and upwind to convection dominated part of the governing equation [1] . Recently higher order finite difference schemes have gained importance due to their interesting properties such as unconditional stability, computational cost, effectiveness and hence efficiency in solving non-linear problems.

The study of recirculation of the fluid inside a square cavity forms the basis to many applications including energy engineering, nuclear reactor [2] , cooling of electronic devices [3] [4] [5] , the study of chaotic mixing [6] , production of plane glass, study of coupling between evaporation and condensation [7] , and in understanding dynamics of water in lakes and ponds [8] . In particular, if the viscosity of the fluid is strongly temperature dependent, then buoyancy effects mix with the inertial effects, leading to complex flow dynamics. In the fluid flow, if the natural buoyancy driven effect and forced shear driven convection effect have comparable magnitude, we have the mixed convective heat transfer. Experimental results on the mixed convection in the bottom-heated rectangular cavity flow show that the heat transfer coefficient is insensitive to the Richardson number [9] . Experimental studies on the natural convection in tilted rectangular cavity have been studied [10] and it is found that the heat transfer depends on the angle of heating the top wall. It is found that for multi-cellular flow is observed which alter the isotherm structure. The instability in the mixed convective flow and heat transfer in a cavity for positive and negative values of Grashof number in which top upper wall is heated with constant temperature are studied [11] and it is found that if the aspect ratio of the cavity is equal to 2, a Hopf bifurcation takes place. A numerical study on the mixed convection lid driven flow in a square cavity with cold vertical walls and sinusoidally heated bottom wall show that the strength of circulation increases with and irrespective of and and further that the overall power law correlation for mean could not be obtained [12] . The effect of different orientation of temperature gradient in the mixed convective heat transfer is studied recently [13] using a finite difference scheme similar to the one in [14] and found that heat transfer rate increases with the decrease of which is independent of the orientation of temperature gradient on the adiabatic walls. It is also found that a thermally stratified fluid will result when the top wall is heated and bottom wall is kept cold. A further extension of studies to evaluate the effect of Richardson and Prandtl number is also reported [15] . Essentially, most of the studies in the literature focus on the flow and heat transfer properties due to bottom uniformly and non-uniformly heated surfaces [13] [15] - [25] , studies emerging due to heating of vertical walls [26] - [33] , reports on uniformly heated top wall [34] [35] , and studies employing internal heat sources [36] [37] . A summary of previous studies employing different numerical schemes with various heating configurations is listed in Table 1. In the present work, we undertake a systematic analysis of mixed convection flow and associated heat transfer effects in a flow induced by a non-uniformly heated top lid which is moving uniformly using a high order accurate numerical scheme coupled with multigrid method.