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A stable high order finite difference with a fully implicit time-marching scheme is proposed to solve the two-dimensional unsteady convection diffusion equation with variable coefficients. It is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions. A time-dependent multigrid full approximation storage(FAS) scheme , which is suitable for both linear and nonlinear problem, is employed to accelerate convergence for the implicit scheme at each time step. Numerical simulations of the unsteady driven flow and natural convection in a square cavity are performed by the present method. The study demonstrates that the method developed here is very accurate, computationally efficient and is capable of performing accurate simulations of time-dependent, and possibly chaotic, flows in enclosures. INTRODUCTION Numerical solution of the convection diffusion equation plays a very important role in computational fluid dynamics and numerical heat transfer to simulate heat and fluid flow problems. Traditional finite difference discretization schemes such as the second-order central difference scheme and the first-order upwind scheme have the drawbacks of either lack of stability (central difference) or lack of accuracy (upwind). Recently, there has been growing interest in developing fourth-order finite difference schemes for the convection diffusion equation (and the Navier-Stokes equations) which give high accuracy approximations (Dennis,1989; Gupta , 1997; Zhang, 1997). Although considerable amount of work has done in the past, researchers are concerned more on steady problems(Dennis,1989; Gupta, 1997; Zhang, 1997) than unsteady ones which are studied by the characteristics difference method (Douglas ,1982 ) and the Group Explicit(GE) method(Evans, 1985). There is still a lack of a completely satisfactory computational (stability and accuracy) scheme that is suitable for all types of unsteady convection diffusion equations. Therefore, in this paper, for the two dimensional unsteady convection diffusion equation with variable coefficients ) , , ( ) ( ) , , ( ) , , ( t y x f a t y x q t y x p yy xx y x t (1) in which, the convection coefficients and are functions of the variables ) , , ( t y x p ) , , ( t y x q y x, and and assumed to be sufficiently smooth. The diffusion coefficient is constant. t a Firstly, a fourth-order compact finite difference in the space directions and a second-order fully implicit time-marching scheme is presented; Then, a time-dependent multigrid FAS scheme is applied to accelerate convergence process when iterative methods are used to treat the implicit scheme at each time step; Finally, driven flow and natural convection in a square cavity as two test problems are simulated to examine accuracy and efficiency of our basic scheme’s applications in the calculation of heat transfer and fluid flow. HIGH ACCURACY DIFFERENCE SCHEME Considering the classical BTCS scheme (Backward for Time and Center for Space), characteristics difference method (Douglas ,1982) and the GE method (Evans,1985),has only low order accuracy in space whereas improving the accuracy of schemes, especially that of convection terms, is very crucial and efficient for improving the stability, accuracy of schemes and decreasing the computational cost(by discretizing computational domain with comparably coarse grids). Therefore, in this paper, we firstly construct a high order compact full implicit difference scheme. Assuming a uniform grid in both xand ydirections, we number the grid points (x,y), (x+h,y), (x,y+h), (x-h,y), (x,y-h), (x+h,y+h), (x-h,y+h), (x-h,y-h) and (x+h,y-h) as 0, 1, 2, 3, 4, 5, 6, 7 and 8 respectively ,where h is the grid size. In writing the finite difference approximations, a single subscript j denotes the corresponding function value at the grid point numbered j. denotes time step size and , j i ij g g g g can be q p , , etc. Following the discretizing method by Ge (2003), a second-order backward Euler difference scheme for the time derivative term and making use of the fourth-order compact difference formula (Hirsh ,1975) for the second order space derivative terms while the first order derivative terms are discretized by center difference scheme but the third order derivative term is kept for modification. Finally we can get its fourth-order full implicit discretization scheme