SUMMARYA class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection-di usion equation with variable convection coe cients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter and fourth order accurate in space. For 0:56 61, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They e ciently capture both transient and steady solutions of linear and nonlinear convectiondi usion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection-di usion problem and three ows of varying complexities governed by the two-dimensional incompressible Navier-Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, e cient and accurate.

SUMMARYA higher order compact (HOC) ÿnite di erence solution procedure has been proposed for the steady twodimensional (2D) convection-di usion equation on non-uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. E ectiveness of the method is seen from the fact that for the ÿrst time, an HOC algorithm on non-uniform grid has been extended to the Navier-Stokes (N-S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for di cult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non-uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex ows such as that in the lid-driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen.

The flow in a thermally driven square cavity with adiabatic top and bottom walls and differentially heated vertical walls for a wide range of Rayleigh numbers (10(3)< or =Ra< or =10(7)) has been computed with a fourth-order accurate higher-order compact scheme, which was used earlier only for the stream-function vorticity (psi-omega) form of the two-dimensional steady-state Navier-Stokes equations. The boundary conditions used are also compact and of identical accuracy. In particular, a compact fourth-order accurate Neumann boundary condition has been developed for temperature at the adiabatic walls. The treatment of the derivative source term is also compact and has been done in such a way as to give fourth-order accuracy and easy assimilation with the solution procedure. As the discretization for the psi-omega formulation, boundary conditions, and source term treatment are all fourth-order accurate, highly accurate solutions are obtained on relatively coarser grids. Unlike other compact solution procedure in literature for this physical configuration, the present method is fully compact and fully higher-order accurate. Also, use of conjugate gradient and hybrid biconjugate gradient stabilized algorithms to solve the symmetric and nonsymmetric algebraic systems at every outer iteration, ensures good convergence behavior of the method even at higher Rayleigh numbers.

In this paper, we propose an implicit higher-order compact (HOC) finite difference scheme for solving the two-dimensional (2D) unsteady Navier-Stokes (N-S) equations on nonuniform space grids. This temporally second-order accurate scheme which requires no transformation from the physical to the computational plane is at least third-order accurate in space, which has been demonstrated with numerical experiments. It efficiently captures both transient and steady-state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. The proposed scheme is likely to be very useful for the computation of transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation. Numerical results are presented and compared with analytical as well as established numerical data. Excellent comparison is obtained in all the cases.