On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system
On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system
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DOI：
10.1051/m2an/2023071
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发表时间：
2023-09
期刊：
影响因子：
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通讯作者：
Yukun Yue;Franziska Weber
中科院分区：
文献类型：
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作者：
Yukun Yue;Franziska Weber
我们提出了基于不变能量二次化方法 (IEQ) 的无条件能量稳定一阶半离散数值格式的收敛分析，该数值格式是为流体动力 Q 张量模型（即所谓的 Beris-Edwards 系统）设计的。该模型由流体流动的纳维-斯托克斯方程组成，与描述液晶分子排列的 Q 张量梯度流耦合。通过使用不变能量二次化方法，我们得到了线性隐式格式，加快了计算速度。然而，这引入了一个辅助变量来代替大量势能，并且先验地不清楚重新表述的系统是否等同于 Beris-Edward 系统。在这项工作中，我们证明了该方案的稳定性，并证明了其收敛于耦合液晶系统的弱解。我们还证明了重构系统和原始系统在弱意义上的等价性。
We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier-Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.