  We present a numerical scheme for approximating the incompressible
Navier-Stokes equations based on an auxiliary variable associated with the
total system energy. By introducing a dynamic equation for the auxiliary
variable and reformulating the Navier-Stokes equations into an equivalent
system, the scheme satisfies a discrete energy stability property in terms of a
modified energy and it allows for an efficient solution algorithm and
implementation. Within each time step, the algorithm involves the computations
of two pressure fields and two velocity fields by solving several de-coupled
individual linear algebraic systems with constant coefficient matrices,
together with the solution of a nonlinear algebraic equation about a {\em
scalar number} involving a negligible cost. A number of numerical experiments
are presented to demonstrate the accuracy and the performance of the presented
algorithm.
