Numerical Approximation of Incompressible Navier-Stokes Equations Based on an Auxiliary Energy Variable

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.