Numerical Approximation of Incompressible Navier-Stokes Equations Based on an Auxiliary Energy Variable
For computational fluid dynamics practitioners, this offers a stable and efficient solver, though it is an incremental improvement over existing energy-stable methods.
The paper introduces a numerical scheme for the incompressible Navier-Stokes equations that ensures discrete energy stability via an auxiliary energy variable, enabling efficient decoupled linear solves and a scalar nonlinear equation. Numerical experiments demonstrate accuracy and performance.
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.