A new class of splitting schemes for incompressible flows is
introduced. The new schemes are based on a weak form of the pressure
Poisson equation and, at each time step, only require to solve a set
of Helmholtz-type equations for the velocity and a Poisson equation
(in the weak form) for the pressure, just as pressure-correction and
velocity-correction schemes.  However, unlike pressure-correction and
velocity-correction schemes, the new splitting schemes are free of
splitting errors and deliver full accuracy on the vorticity and the
pressure.