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A stability condition is provided for a class of vorticity boundary formulas used with the second order finite-difference numerical scheme for the vorticity-stream function formulation of the unsteady incompressible Navier-Stokes equations. These local vorticity boundary formulas are derived using the no-slip boundary condition for the velocity. A new form of these long-stencil formulas is needed to classify the stability property, in which local terms are controlled by global quantities via discrete elliptic regularity for the stream functions.

A General Stability Condition for Multi-stage Vorticity Boundary Conditions in Incompressible Fluids