Numerical schemes are presented for the incompressible Navier-Stokes
equations based on a primitive variable formulation in which pressure
is treated explicitly in time and the incompressibility constraint has
been replaced by a pressure Poisson equation. The crucial point for
these schemes is the proper enforcement of a Neumann boundary
condition for the pressure Poisson equation, which in turn ensures
satisfaction of the divergence-free condition for the velocity field.
The computation of the momentum and kinematic equations are fully
decoupled, resulting is a class of extremely efficient Navier-Stokes
solvers. Moreover, the schemes are not projection-type methods and are
free of numerical boundary layers resulting from time consistency
issues inherent in such splitting methods. Full time accuracy is
achieved for all flow variables in the L∞
norm. In particular, unconditional stability of this approach applied
to the Stokes's equation is proven for both first or second order
implicit time discretization. Numerical examples are presented using
finite differences, collocation and Galerkin spectral methods,
demonstrating the flexibility of the approach in the choice of spatial
discretization.
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