An upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier-Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction-velocity formulation of the unsteady incompressible Navier-Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both (n)th and (n+1)th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction-velocity formulation in the aspect of computational cost.