We treat the streamfunction formulation of the Navier-Stokes equations.
The differential equation and boundary and initial conditions are formulated via the streamfunction only.
A compact scheme, which invokes the streamfunction and its gradient at mesh points, is introduced for the streamfunction formulation.
Only the eight nearest neighboring points are involved in the discretization.
We demonstrate our results on physical problems, such as the driven cavity problem and for image inpainting as well.
The convergence of the scheme is proven for the full non-linear Navier-Stokes equations.
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