A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows

In this paper, we extend a previous work on a compact scheme for the steady Navier-Stokes equations [Li, Tang, and Fornberg (1995), Int. J. Numer. Methods Fluids, 20, 1137-1151] to the unsteady case. By exploiting the coupling relation between the streamfunction and vorticity equations, the Navier-Stokes equations are discretized in space within a 3 x 3 stencil such that a fourth order accuracy is achieved. The time derivatives are discretized in such a way as to maintain the compactness of the stencil. We explore several known time-stepping approaches including second-order BDF method, fourth-order BDF method and the Crank-Nicolson method. Numerical solutions are obtained for the driven cavity problem and are compared with solutions available in the literature. For large values of the Reynolds number, it is found that high-order time discretizations outperform the low-order ones.