Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable

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 scalar number involving a negligible cost. A number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm.

A gPAV-based unconditionally energy-stable scheme for incompressible flows with outflow/open boundaries

“…• The dynamic equations for the auxiliary variables, and their numerical discretizations, in the current work and in [37] are completely different. In [37] a nonlinear algebraic equation needs to be solved based on the Newton's method when computing the auxiliary variable. In contrast, the auxiliary variable in the current work is computed by a well-defined explicit formula, and no nonlinear algebraic solver is involved.…”

“…In contrast, the auxiliary variable in the current work is computed by a well-defined explicit formula, and no nonlinear algebraic solver is involved. Furthermore, the computed values for the auxiliary variable here are guaranteed to be positive, and this positivity property is unavailable in the method of [37]. These points will become clear in subsequent discussions.…”

“…Remark 2.3. Note that the numerical scheme from [37] also employs an auxiliary variable. Several major differences distinguish the scheme herein from the one from [37]:…”

“…In the implementation in [37], a further approximation is made to decouple the computations for the pressure and the velocity. The discrete energy stability of [37], however, breaks down mathematically with that approximation. In contrast, in the current scheme the pressure and the velocity are de-coupled, except for the g(ξ) term, which can be dealt with in a straightforward way (see later discussions).…”

“…The adoption of the auxiliary variable enables us to deal with the ESOBC in a relatively simple way. It should be noted that the auxiliary variable and the Navier-Stokes equations are treated in a very different way in the current work than in [37]. In this work the incompressible Navier-Stokes equations, the dynamic equation for the auxiliary variable, and the energy-stable open boundary conditions have been reformulated based on the generalized Positive Auxiliary Variable (gPAV) approach.…”