An Energy-Stable Scheme for Incompressible Navier-Stokes Equations with Periodically Updated Coefficient Matrix

TL;DR

This paper introduces an energy-stable numerical scheme for incompressible Navier-Stokes equations that uses periodically updated coefficient matrices to improve computational efficiency while maintaining accuracy and stability.

Contribution

The novel scheme combines the gPAV framework with periodic coefficient matrix updates, enabling efficient and stable simulations of Navier-Stokes equations with large time steps.

Findings

The scheme maintains energy stability with large time steps.

Periodic updates of the velocity coefficient matrix have minimal impact on computational cost.

Numerical experiments demonstrate the method's accuracy and efficiency.

Abstract

We present an energy-stable scheme for simulating the incompressible Navier-Stokes equations based on the generalized Positive Auxiliary Variable (gPAV) framework. In the gPAV-reformulated system the original nonlinear term is replaced by a linear term plus a correction term, where the correction term is put under control by an auxiliary variable. The proposed scheme incorporates a pressure-correction type strategy into the gPAV procedure, and it satisfies a discrete energy stability property. The scheme entails the computation of two copies of the velocity and pressure within a time step, by solving an individual de-coupled linear equation for each of these field variables. Upon discretization the pressure linear system involves a constant coefficient matrix that can be pre-computed, while the velocity linear system involves a coefficient matrix that is updated periodically, once every $k_0$ time steps in the current work, where $k_0$ is a user-specified integer. The auxiliary variable, being a scalar-valued number, is computed by a well-defined explicit formula, which guarantees the positivity of its computed values. It is observed that the current method can produce accurate simulation results at large (or fairly large) time step sizes for the incompressible Navier-Stokes equations. The impact of the periodic coefficient-matrix update on the overall cost of the method is observed to be small in typical numerical simulations. Several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein.