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

TL;DR

This paper introduces a novel energy-based numerical scheme for the incompressible Navier-Stokes equations that ensures discrete energy stability and enables efficient computation through decoupled linear systems.

Contribution

The paper proposes an auxiliary energy variable approach that reformulates the Navier-Stokes equations, leading to a stable and efficient numerical scheme with a simplified nonlinear solve.

Findings

The scheme is energy stable in a discrete sense.

Numerical experiments confirm accuracy and efficiency.

The method simplifies the solution process with decoupled linear systems.

Abstract

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.