A new family of (5,5)CC-4OC schemes applicable for unsteady Navier–Stokes equations

•A new family of implicit spatially fourth order accurate compact finite difference scheme has been proposed.•The system of Navier–Stokes equations reduces to a diagonally dominant constant coefficients algebraic system.•The family of schemes enjoys better resolution properties compared to other compact formulations.•Linear stability analysis shows the schemes are unconditionally stable.•The extension of the proposed discretization procedure to irregular domains and three dimensional cases has been discussed.

In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection–diffusion equation with variable convection coefficient is proposed. The schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady Navier–Stokes system. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Padé scheme for spatial derivative. These schemes which are based on a five point stencil with constant coefficients, named as “(5,5) Constant Coefficient 4th Order Compact” [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary conditions. Subsequently the proposed schemes are applied to problems governed by the incompressible Navier–Stokes equations. The results obtained are in excellent agreement with analytical and available numerical results in all cases, establishing efficiency and accuracy of the proposed schemes.