We consider two-dimensional unsteady Rayleigh-Benard convective motion of a viscoelastic fluid in a square cavity. The governing vorticity and energy transport equations are discretised by using finite difference approximations and solved numerically by either simple explicit or ADI methods, respectively. The two-dimensional convective motion is generated by buoyancy forces on the fluid in a square cavity, when the horizontal walls are maintained at two constant temperatures and the vertical walls are perfectly insulated. In this spirit we consider Rayleigh-Benard convection heat transfer in a square cavity shear-thinning and elastic effects (the first normal stress difference), as embodied in the dimensionless shear-thinning and elastic numbers, have an influence in shaping the flow field and determining the heat transfer characteristics with respect to the Rayleigh numbers. Their combined effect acts to increase and decrease the heat transfer as represented by the local Nusselt number with the results that overall heat transfer for the viscoinelastic and viscoelastic cases and all results in this paper have been documented first time for the viscoelastic fluid. (C) 2002 Elsevier Science Inc. All rights reserved.