Journal of Computational Physics ( IF 4.1 ) Pub Date : 2020-06-02 , DOI: 10.1016/j.jcp.2020.109624 Lianlei Lin , Naxian Ni , Zhiguo Yang , Suchuan Dong
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 time steps in the current work, where 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.
中文翻译:
具有周期更新系数矩阵的不可压缩Navier-Stokes方程的能量稳定方案。
我们提出了一种基于广义正辅助变量(gPAV)框架的不可压缩Navier-Stokes方程的能量稳定方案。在gPAV重组系统中,原始非线性项由线性项和校正项代替,其中校正项受辅助变量控制。拟议的方案将压力校正类型的策略纳入gPAV程序,并且满足离散的能量稳定性能。该方案需要通过为每个场变量求解独立的解耦线性方程,在一个时间步内计算速度和压力的两个副本。离散化后,压力线性系统会包含一个可以预先计算的常数系数矩阵, 当前工作中的时间步骤 是用户指定的整数。辅助变量是一个标量值的数字,由定义明确的显式公式计算,该公式可保证其计算值的正性。可以观察到,对于不可压缩的Navier-Stokes方程,当前方法可以在较大(或相当大)的时间步长上产生准确的模拟结果。在典型的数值模拟中,观察到周期性系数矩阵更新对方法总成本的影响很小。已经模拟了几个流动问题,以证明本文开发的方法的准确性和性能。