An energy-stable scheme for incompressible Navier-Stokes equations with periodically updated coefficient matrix

JOURNAL OF COMPUTATIONAL PHYSICS

卷:418

Article

Lin, Lianlei¹  Ni, Naxian²  Yang, Zhiguo²  Dong, Suchuan² 

[1] Harbin Inst Technol, Sch Elect & Informat Engn, Harbin, Heilongjiang, Peoples R China

[2] Purdue Univ, Dept Math, Ctr Computat & Appl Math, W Lafayette, IN 47907 USA

关键词: Energy stability;    Navier-Stokes equations;    Incompressible flows;    Auxiliary variable;    Generalized positive auxiliary variable;    Pressure correction;   

DOI  :  10.1016/j.jcp.2020.109624

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier Inc. All rights reserved.

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