不可压外流/开边界问题的gPAV-Based无条件能量稳定格式
A gPAV-Based Unconditionally Energy-Stable Scheme for Incompressible Flows with Outflow/Open Boundaries
摘要：
本文提出了一种无条件能量稳定的格式，用于求解外流/开边界区域上的不可压纳维-斯托克斯方程。该格式结合了广义正辅助变量（gPAV）方法和转速修正型策略，辅助变量的引入简化了开边界条件的数值处理。该方案的离散能量稳定性已被证明，无论时间步长大小。在每个时间步长内，该计划需要计算两个速度场和两个压力场，通过求解一个单独的解耦亥姆霍兹（包括泊松）型方程与一个常数的可预先计算的系数矩阵，这些字段变量。辅助变量是一个标量，在一个时间步长内由一个明确定义的显式公式给出，这确保了其计算值的正性。大量的数值实验与几个流动涉及流出/开放边界的制度，回流不稳定性变得严重已被提出来测试所提出的方法的性能，并证明其在大的时间步长的稳定性。
英文摘要：
We present an unconditionally energy-stable scheme for approximating the incompressible Navier-Stokes equations on domains with outflow/open boundaries. The scheme combines the generalized Positive Auxiliary Variable (gPAV) approach and a rotational velocity-correction type strategy, and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions. The discrete energy stability of the proposed scheme has been proven, irrespective of the time step sizes. Within each time step the scheme entails the computation of two velocity fields and two pressure fields, by solving an individual de-coupled Helmholtz (including Poisson) type equation with a constant pre-computable coefficient matrix for each of these field variables. The auxiliary variable, being a scalar number, is given by a well-defined explicit formula within a time step, which ensures the positivity of its computed values. Extensive numerical experiments with several flows involving outflow/open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes.
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