An efficient compact difference scheme for solving the streamfunction formulation of the incompressible Navier-Stokes equations

Recently, a new paradigm for solving the steady Navier-Stokes equations using a streamfunction-velocity formulation was proposed by Gupta and Kalita [M.M. Gupta, J.C. Kalita, A new paradigm for solving Navier-Stokes equations: streamfunction-velocity formulation, J. Comput. Phys. 207 (2005) 52-68], which avoids difficulties inherent in the conventional streamfunction-vorticity and primitive variable formulations. It is discovered that this formulation can reached second-order accurate and obtained accuracy solutions with little additional cost for a couple of fluid flow problems. In this paper, an efficient compact finite difference approximation, named as five point constant coefficient second-order compact (5PCC-SOC) scheme, is proposed for the streamfunction formulation of the steady incompressible Navier-Stokes equations, in which the grid values of the streamfunction and the values of its first derivatives (velocities) are carried as the unknown variables. The derivation approach is simple and can be easily used to derive compact difference schemes for other similar high order elliptic differential equations. Numerical examples, including the lid driven cavity flow problem and a problem of flow in a rectangular cavity with the hight-width ratio of 2, are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The results obtained are compared with ones by different available numerical methods in the literature. The present scheme not only shows second-order accurate, but also proves more effective than the existing second-order compact scheme of the streamfunction formulation in the aspect of computational cost.

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